Average Rate of Change Over Interval

Welcome MathSux friends! In todays lesson, we will be going over how to find the average rate of change over an interval of a function. This may sound intimidating at first, but all we are really doing is finding the slope, yes, the one we’re already familiar with over an interval given usually in the x-axis. Sounds simple enough? Check out the example below! Also, be sure to watch the video to check out multiple ways for solving these types of problems and the practice problems at the end of this post. Happy calculating!

Average rate of Change over Interval Example:

Step 1:  The first thing we must do is figure out what this question is saying. The interval, they give us, [1,3] represents the x-values on our graph.  So, they want to know what the average rate of change (or slope) is between these two x-values 1 and 3.  Let’s take a look and see what that means for us on the graph below!

Average Rate of Change Over Interval

Notice we found the points on our interval [1,3]?  This correlates to the coordinate points we will be working with to solve this problem (1,2) and (3,10)

Also, within our interval, [1,3], we will be finding the rate of change (otherwise known as the slope) of the two coordinates (1,2) and (3,10). Check out how we are only finding a small portion of our slope from this function in the diagram below, represented by the dotted green line.

Average Rate of Change Over Interval

Step 2:  Now that we understand what we must do, find the slope within the interval [1,3], using the corresponding points, (1,2) and (3,10), all we must do is plug our numbers into the good ole’ slope formula which we already know!

Using our coordinates, (1,2) and (3,10), we can plug in values into the slope formula below:

Try the following questions on your own on and check out the video above for more ways to answer this type of question and even more examples!

Practice Questions:

1) Find the average rate of change over the interval [0,2] for the function: h(x) = 2x2 +2

2) Find the average rate of change over the interval [-4,2] for the function: g(x)=(x+3)2

3) Find the average rate of change over the interval [-2,0] for the function: f(x)=-(x+1)3

4) Find the average rate of change over the interval [0,4] for the function: h(x)=x2+2x+1

Solution:

1) 4

2) 1

3) -1

4) 6

Looking to learn more about Algebra? Check out the algebra lessons page here. Thanks for stopping by and happy calculating! 🙂

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What is Trigonometry

What is Trigonometry?

What is Trigonometry? Trigonometry is the study of triangles angles and sides in mathematics. By applying the rules of trigonometry, we can find unknown angles and side lengths in triangles and other shapes that can be broken down into triangles.

Who cares? Why do we need to know a triangles angles or side length? This is a fair question! Although on paper, trigonometry can feel useless, it is a great tool in solving real-world problems in architecture, astronomy, engineering, and even video game design! For example, what if we wanted to know the angle measure between our line of sight and the distance to the moon? Or what if wanted to find the perfect angle for a roof on a building? All of these questions would be answered using the beautiful subject of trigonometry.

Just like the word triangle, trigonometry includes the prefix “tri” meaning three. Three, the magic number in trigonometry as there also happens to be three main trigonometric functions (sine, cosine, and tangent) along with their inverses (secant, cosecant, and cotangent respectively). Sine, cosine, and tangent are the basic trig functions that allow us to find the values of angles and/or sides of a triangle of a right triangle.

History of Trigonometry

Where did trigonometry come from? The word trigonometry can be traced back to Greece, from the greek word “trignon” meaning triangle and the word “metron” which means to measure. Clearly trigonometry has history in Greece dating back to the 3rd century B.C. Although it has roots in Greek history, the subject can also be traced back to India in 5th century A.D. Each culture independently inventing trigonometry at first for astronomical purposes.

Basic Right Triangle Trigonometry

We have briefly touched upon the basic trigonometric functions (sine, cosine, and tangent), now let’s dive a little deeper and see how each function works with a right triangle!

Sine, Cosine, and Tangent, fondly known as Sin, Cos, and Tan are trigonometric functions that can be used to find angles and sides of right triangles (triangles with a 90º angle). Sin, Cos and Tan can be summarized by the very memorable acronym SOH CAH TOA:

what is trigonometry

The greek letter, θ, pronounced “theta,” is used to represent unknown angles in trigonometry and can be paired with each trig ratio (sin θ, cos θ, tan θ) which is exactly wha you see above.

In order to use our trig functions, we need to define the different parts of a right triangle based on the location of the angle. There will always be an opposite side that is opposite to angle θ, an adjacent side that is next to angle θ. And the last remaining side, the only side that will always be labeled the same (regardless of the angle θ) is the longest side of a right triangle, which is called the hypotenuse.

Let’s try an Example:

Given the following right triangle, find sin30º .

Step 1: Since we are going to find the sin30º, let’s write out our ratio for sine based on our acronym SOH CAH TOA. 

Step 2: Now we must identify our opposite and hypotenuse based on our angle θ. Remember the opposite is the opposite side length in relation to our angle θ, which in this case is 1.

The hypotenuse is always the longest side, which in this case is 2.

(The adjacent leg is radical 3, but in the case of finding sine, we don’t need this value).

acute angle
opposite leg

Step 3: Now that we have our values for the opposite (1) and hypotenuse (2), let’s plug them into our sin ratio for for our answer.

trigonometric ratio

If you are looking for more right triangle and SOH CAH TOA practice questions similar to the one above, please check out the video and link here.

Inverse Trigonometric Functions:

Inverse Trigonometric ratios are the inverses of the regular trig functions (sin, cos, and tan) we just went over! Check it out below:

inverse trigonometric function

We solve inverse trigonometric functions, the same exact way we do regular trig functions, let’s take a look at an example below:

Find the exact value of csc45º

right angle

Step 1: We know thatCSC is the inverse of sin, which is equal to 1/sin , so let’s use some SOH CAH TOA action to solve for sin45º as normal, then flip our answer to find the value of csc45º.

trigonometric function

Step 2: Now all we need to do is “flip” the answer we got in Step 1 to get its inverse which will be the value of csc45º.

trigonometric function

Unit Circle:

The Unit Circle is an important reference tool used throughout trigonometry to derive all sorts of formulas and explains how trig works in the first place!

Why is it called a unit circle?

Notice each point on the circle is one unit away from the origin below.

The Unit Circle

Why do the degrees go counter-clockwise?

A circle has  360º therefore, we can re-label the axis with 0º, 90º, 180º, 270º, and 360º . Notice the numbers go in a counterclockwise direction? This is based on the movement of the Earth around the sun, as it too travels counterclockwise.  Fun fact: Astronomy and the study of the movement of the sun and the earth are the origins of trigonometry!

Other things to notice about the Unit Circle:

1. Degrees follow a counter-clockwise pattern from 0 to 360 degrees.
2. Values of cosine are represented by x-coordinates.
3. Values of sine are represented by y-coordinates.
4. Using the unit circle we can find the degree and radian value of trigonometric functions (SOH CAH TOA).

Trig Functions and Quadrants:

Based on finding angles via the Pythagorean Theorem using reference triangles within each quadrant we can identify, to see which trig functions (sin, cos, tan) are positive in which quadrant. Let’s see how step by step with a proof!

Step 1: To prove why certain trig functions are positive in specific quadrants, draw a unit circle and add triangles within each quadrant. Notice it kind of looks like a bow tie and we added an angle θ within each triangle as well, this will come in handy later!

Step 2: Now we are going to find the trigonometric functions (sine, cosine, tangent) with respect to  θ for each triangle in each quadrant. Let’s close up on our triangle in quadrant I.

what is trigonometry

Notice the hypotenuse has a value of 1 because this is a unit circle, and “1” is the value of the radius.

We do not know the x and y values for this triangle, but we do know that both  x and y will be positive values.

If we were to find sinθ, we would get:

right angled triangle
adjacent side

What if we found cosθ and tanθ for our Quadrant I reference triangle?

We just proved that all our trig functions are positive in Quadrant I!  Check  below to see about Quadrants II, III, and IV:

Things to notice:

Notice all the positive functions are hi-lighted in green.

Just like we proved Quadrant I to be positive for all trig functions, the above diagram shows that only sin is positive in QII, only tan is positive in QIII and only cos in QIV.

We derived each, by using SOH CAH TOA on each reference right triangle with respect to θ.

There is much more to be discovered about the unit circle, if you are interested in learning more, please check out the video below and link here

Graphing Trig Functions:

We’ve seen the unit circle, now I have to wonder, what do trig functions (sin, cos, tan) look like when they are graphed on a coordinate plane? Get ready because they look pretty cool and the visualizations of these trig functions may surprise you!

Have you ever heard of a sine wave? A radio wave? That’s exactly what trig functions graphs look like because that is what they are!

Sin Graph: Sine function curves creates what looks like an “S” shape. As they say, “S” is for Sine, this is the easiest way to remember what the sine function looks like, check it out below!

Graphing Trig Functions

Cosine Graph: While Graphing Cosine, you may notice that the cosine function creates what looks like a “V” shape. As they say, “V” is for victory, but of course, in this case, it is for cosine!

trigonometry function

Tangent Graph: The tangent function looks totally different graphed when compared to sine and cosine. You may notice the dotted vertical lines below, these are called asymptotes and no values can exist here, check it out below!

Graphing Trig Functions

Why do trigonometric function graphs look this way? Derive and learn why trig ratios graphed look unique and more here. And if you want to learn how to transform a trig graph, check out this link here.

Special Triangles:

Special triangles, otherwise known as the superheroes of trigonometry, are a valuable reference tool for solving trigonometric functions.  The special triangles include the 30 60 90 triangle and the 45 45 90 triangle. Each of the right triangles you are about to see can be derived from the unit circle.

By knowing the below special triangles, we can answer questions such as “find the value of sin45º  and cos45º  without using a calculator.”  Special triangles will save the day when it comes to finding the unknown values of angles in a triangle.  Now, let’s look at our very special triangles below:

Trigonometric Identity:

Throughout trigonometry, there are many trigonometric identities that are derived which help solve proofs as well as make factoring linear/quadratic trig equations easier. Below is a sample of the trigonometric identities you will come across in this subject:

Inverse Identities:

Trig Identities
Inverse Function

Ratio Identities:

Trig Identities

Pythagorean Identity:

Trig Identities
what is trigonometry

If you are looking for more information on trigonometric identities, please check out the video below and practice questions here!

Finding Angles and Sides in Non-Right Triangles:

How would we find an unknown angle or side of a triangle, when the triangle is NOT a right triangle? The answer is to use the law of sines or the law of cosines. How do I know when to use the law of sines versus the law of cosines? It all depends on the information of the triangle that is given to us of course! See the differences below:

Law of Sines: Use when given ASA, AAS, and ambiguous case SSA of a triangle.

law of sines what is trigonometry

Law of Cosines: Use when given SSS or SAS of a triangle.

a2 = b2 + c2 -2bcCosA

b2 = a2 + c2 -2acCosB

c2 = a2 + b2 -2abCosC

Still got questions? Have a favorite part of trigonometry you want to share? Don’t hesitate to email or comment below! Also, don’t forget to follow us on social media to get the latest and greatest MathSux lessons, videos, questions, and more. Happy Calculating!

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Law of Cosines

Greeting math friends and welcome to MathSux! In today’s post, we are going to cover the Law of Cosines!  Otherwise known as the cosine rule, this is a great formula for finding missing angle and side information on a triangle that is NOT a right triangle when we have SAS or SSS information about the triangle in question.  This formula is much more straight forward when compared to the law of sines, so if you have already mastered that formula, the law of cosines should be a breeze! Also, don’t forget to check out the practice questions at the end of this post to test your knowledge of the law of cosines.  Happy calculating!

Law of Cosines:

The law of cosines allows us to find missing sides and angles of any non-right triangle (a.k.a an oblique triangle) where we are given SAS or SSS information about the triangle in question.

If you take a peak below at the law of cosines, notice that it can be derived from the Pythagorean theorem, as it is reminiscent of the classic formula we all know and love, a2 + b2 = c2.

Also, notice each angle is across its opposite side (Angle A is opposite side a, Angle B is opposite to side b, and Angle C is opposite to side C).

a2 = b2 + c2 -2bcCosA

b2 = a2 + c2 -2acCosB

c2 = a2 + b2 -2abCosC

To find Angles we can also derive and use the following formulas based on the law of cosines from above:

law of cosines

Example #1: SAS Find the value of the missing side.

We can use the law of cosines to find missing sides when we are given an oblique triangle that has a side, angle, and side (SAS) information.

law of cosines

Step 1: First, let’s identify what type of information our triangle is providing us with.  Notice that we are given a side, an angle, and a side for SAS. Since our triangle contains SAS, we know we can apply the cosine rule to find the value of unknown side c.

SAS

Step2: Next, let’s write out our law of cosines formula to find the value of length c and plug in our given information to solve for c. In this case we can plug in for side a=10, side b=8, and opposite side c we have angle c=40º.

Remember to take the square root of both sides of the equation to get our solution for unknown side c!

law of cosines
acute triangle

Example #2: SSS Find the value of the missing angle.

When we want to find the value of a missing angle and are given SSS information of a triangle, we can either use the original law of cosines formula and use our knowledge of algebra and trigonometry to solve for the unknown angle, or, we can also a different version of the cosine rule shown below.  Either method works and will give you the correct answer, the choice is up to you!

law of cosine angles
law of cosines

Step 1: First, let’s identify what type of information our triangle is providing us with.  Notice that we are given a side, side, and another side for SSS. Since our triangle contains SSS, we know we can apply the cosine rule to find the value missing angle C.

SSS

Step 2: Next, let’s write our law of cosines formula and then plug in our given information. In this example, I will be using the original formula and then using algebra and trigonometry to find the correct answer, but feel free to apply one of the angle formulas above and see if you get the same answer!

Since we want to find the value of angle C, we will be using the formula that is set equal to C2:

 cosine
cosine

Think you are ready for more!? Try answering the following practice questions on the cosine rule on your own to truly master this formula! Then check your answers with the solutions below.

Practice Questions:

1) Find the value of missing side c, to the nearest hundredth.

2) Find the value of missing side a, to the nearest hundredth using the cosine formula.

obtuse triangle

3) Find the angle measure of unknown angle A, to the nearest degree using the cosine rule.

4) Find the value of missing angle B, to the nearest hundredth using the cosine rule.

Solution:

Still got questions on the law of cosines? Would you like to see how the cosine rule can be derived from the pythagorean theorem? Don’t hesitate to comment with any questions! If you want to check answers on your homework, you can also check out the law of cosines calculator here! And if you’re looking for more on trigonometry check out the related posts below.Happy calculating! 🙂

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Related Trigonometry Posts:

The Unit Circle

Basic Right Triangle Trigonometric Ratios (SOH CAH TOA)

4545 90 Special Triangles

30 60 90 Special Triangles

Graphing Trigonometric Functions

Transforming Trig Functions

Factoring Trigonometric Functions

Trig Identities

Law of Sines

Greeting math friends and welcome to MathSux! In today’s post, we are going to go over the Law of Sines!  This is a great formula for finding missing angle and side information on a triangle that is NOT a right triangle, that is only if we are given ASA (Angle, Side, Angle) or AAS (Angle, Angle, Side) information of said triangle. 

There is also something called the “ambiguous case” that happens when we have SSA (Side, Side, Angle) information of a triangle, and it is just as mysterious as it sounds.  We can use the law of sine in the case of SSA to find missing angle and side information of a triangle as well, but there may be 0 triangles, 1 triangle, or even 2 triangles that exists in this case!

Sound confusing? Fear not, because we will cover everything in this post, including practice questions found throughout and at the bottom of this article.  Happy calculating!

Law of Sines:

The law of sines tells us that in any triangle (non-right triangles included!), the ratio of a sin of an angle to the value of its corresponding side are the same for all three sides of a triangle. Check out the formula below:

law of sines

Example #1: AAS Find the value of the missing side

law of sines

Step 1: First, let’s identify what type of information our triangle is providing us with.  Notice that we are given angle A , angle B, and side b for AAS. Since our triangle contains AAS, we know we can apply the law of sine to find the missing side.

sin b 
AAS
b c

Step 2: Next, let’s write our law of sine formula and then plug in our given information. Notice we set up our formula where each side (a and b) and the sine of its opposite angle (Sin A and Sin B) represent the denominator and numerator respectively.

sine rule
sin b

Step 3: Time to use some of our basic algebra and trigonometry skills and solve for unknown side length a by cross multiplying.

sine

* Tip! Remember when plugging sin, sin-1, and all other trigonometric functions into your calculator, to be sure that you are in degree mode!

Example #2: ASA Find the value of the missing side

sin

Step 1: First, let’s identify what type of information our triangle is providing us with.  Notice that we are given an angle A, side c , and an angle B for ASA. Since our triangle contains ASA, we know we can apply the law of sine to find the missing side.

angle sides angle

Step 2: Taking a look at the angles of our triangle, notice how side a is in between angle B and angle C. We have angle B, but notice the value of angle C is missing! So we first will need to find angle C. We can find the value of angle C, knowing that the interior angles of a triangle always add to 180º.

Angle C: 180º – (40º +60º) = 80º

sin

Step 3: Now that we have that missing angle value, we can find the length of side a. Let’s write our law of sines formula and then plug in our given information. Then we can cross multiply and use our knowledge of algebra and trigonometry to find the correct answer.

sine rule

* Tip! Remember when plugging sin, sin-1, and all other trigonometric functions into your calculator, to be sure that you are in degree mode!

Think you are ready to master the law of sines on your own for AAS and ASA triangles? Try the following practice questions on your own to test your law of sines knowledge!

Practice Questions:

1) Find the value of missing side, b, to the nearest tenth.

2) Find the value of missing side, c, to the nearest tenth.

angles a and b

3) Find the value of missing side, b, to the nearest tenth.

4) Find the value of missing side, c, to the nearest tenth.

Solutions:

The Ambiguous Case – SSA and Law of Sines

So far, we’ve seen how to find a missing side using the law of sine when given ASA or AAS of a triangle.  But now, we will see a special case scenario, otherwise known as the Ambiguous Case, where we are given SSA (Side, Side, Angle) of a triangle and must use the law of sines to see if there are potentially 0, 1, or 2 triangles that can potentially exist..

The Ambiguous case, is “ambiguous” because SSA is not enough information to find unknown angles and sides, as there can be more than one possible triangle with different angle measures.  Although there is not enough information, SSA does tell us enough to infer all potential triangles (0,1 or 2) by figuring out what potential angles may or may not exist.

There are a lot of rules and visualizations associated with the ambiguous case that I will not be getting into here.  If you would like to see more information and visualizations of the ambiguous case, please let me know in the comments.  Let’s see how it works with the next example:

Ambiguous Case Example

Step 1: First, let us draw a triangle and fill in the information that has been provided to us, knowing angle A is 30º, side a is 10 and side b has a length of 15.

Step 2: Next, let’s identify what type of information our triangle is providing us with.  Notice that we are given a side, side, and an angle for SSA. Since our triangle contains SSA, we know we can apply the law of sines and that this is going to be an ambiguous case, meaning there can be zero, one, or two potential triangles.

Step 3: Now, we can start off by using the law of sine to find the value of angle b.  We want to find angle B because we already have the value of side length b, opposite our unknown angle B.

sine rule

* Tip! Remember when plugging sin, sin-1, and all other trigonometric functions into your calculator, to be sure that you are in degree mode!

Step 4: But wait! We have found what appears to be value of unknown angle B, but since this is a SSA, ambiguous case, we are not 100% sure of our angle measure because of too many unknowns!

Why can there be more than one value for angle b?

Since we are given SSA of our triangle, that leaves too many unknown values not just for angle B, but also for unknown values angle C, and the length of AB.  This leads us to having more than one possible angle value for angle B. 

Does one triangle exist?  Yes!

We can see that one triangle exists using our found angle, angle b=49º.  Based on this, we can infer what angle C would be, knowing that the interior angles of a triangle add to 180º.

Angle C= 180º – (49º+30º) = 101º

Do two Triangles exist? Yes!

Since our angle B technically can have two values, we can once again look to the unit circle and remember that sin is positive in quadrant 2 (sin must be positive since we are dealing with the law of sines and triangles).  Knowing this we can use our reference angle 180º – θ, to find the potential bonus value for angle b, plugging in the angle value we found previously for angle B= 49º for θ. 

Alternate value for angle B = 180º – θ = 180º – 49º = 131º

obtuse angle

Based on this, we can infer what angle c would be, knowing that the interior angles of a triangle add to 180º.

Alternate value for angle C = 180º – (30º + 131º) = 19º

Note! In this case there are two triangles, but please note that that will not always be the case and the way to know that is to make sure all the angles of your second triangle add up to 180º and nothing over!

Think you are ready to give the ambiguous case and the sine rule a spin? Try the following practice questions on your own!

Practice Questions:

How many triangles can be constructed with the given measures?

law of sines

Solutions:

  1. 2 Triangles
  2. 1 Triangle
  3. 2 Triangles
  4. 1 Triangle

Still got questions on the law of sine? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! If you want to check answers on your homework, check out the law of sines calculator here! 🙂

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Related Trigonometry Posts:

The Unit Circle

Basic Right Triangle Trigonometric Ratios (SOH CAH TOA)

4545 90 Special Triangles

30 60 90 Special Triangles

Graphing Trig Functions

Transforming Trig Functions

Factoring Trig Functions

Law of Cosines

Trig Identities

Best books on History of Mathematics

So many books, such little time! I had this dangerous idea of compiling all the best books found in the mathematical universe. The history of mathematics and the evolution of this global subject spans language barriers, across time, and throughout the world!

Before you discount math as a dry subject devoid of anything interesting, remember that there are people and stories behind each and every mathematical discovery.

Remember that the history of mathematics includes everything you can think of, starting at the beginning with the origins of mathematics and the development of the number zero all the way to the more modern origins of STEM and computers!

The applications of math and logic can also be applied to so many different subjects (science, engineering, technology, chemistry, physics, art, etc.) that mathematical history can also be found in within the stories of these subjects as well.

The history of mathematics is more than just the math itself as it includes a history of people, culture and motivation as to why we as humans needed math in the first place!

The below list of math history books will take us on adventures from ancient civilizations all the way to more modern day stories and movies such as Hidden Figures.

The top math books are broken up by three main categories:

1) Top math history books – Learn in timeline order mathematic concepts and philosophies how and where they came from

2) Biographies – Learn in detail about the people behind the subject of mathematics

2) Top History math books for children- Entertaining for all, but mostly written for children, this collection of books will introduce mathematics as light, interesting, and fun!

Mathematical History Books

Concise History:

The first section is for anyone who wants to dive deep into the history of mathematics on an academic level. Great for the math nerds who want to know the story and history of mathematics, this section is great for reference as a teacher, professor , or just great for anyone who wants to learn more about mathematics and the people behind it.

1) A History of Mathematics by Carl Boyer and Uta C. Merzbach. Explore the history of mathematics through ancient civilizations around the world and connect them to the modern theorems we are all more familiar with. This book will show you the origins of the classroom mathematics we all do and forget why we do in the first place! Though, fair warning, this is a dense book and can read more like textbook, but great information and great source for reference or if you’re looking to read something a bit intense!

mathematical experience

2) Unknown Quantity: A Real and Imaginary History of Algebra by John Derbyshire. Discover the stories and histories behind the subjected of algebra, spanning place and time! See the origins and development of mathematical thought from the past all the way to modern times. In this math book, you will meet many mathematicians, learn about their contributions to the subject, as well as, their stories. If you ever wanted to answer questions such as, why are we learning this in the first place? and where did the quadratic equations come from, this is the math book for you!

mathematical association

3) Men of Mathematics by Eric Temple Bell. This is a mathematic classic (yet maybe dated) history of mathematics book. I only say it may be dated because it was written in 1937. Although it is from the 30’s, we must all admit that the ancient history of math to more modern day nineteenth century calculus with Isaac Newton has not changed and that is why it is still on this list. Just be ready to not hear much in terms of the development of technology, but be ready to learn about achievement in mathematics from several famous mathematicians throughout the ages.

Biography:

Mathematics is an invented subject. Invented by people throughout time all over the world. Learn in detail about who these people were, their struggles, their discoveries, and of course about their contributions to mathematics.

1) Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race by Margot Lee Shetterly. The movie we all know and love was a book first! Always fun to read the book if you haven’t already to see any differences. And for those of you who don’t know what the book is about, it is based on the true story and follows a group of black women working at NASA as mathematicians, known as “human computers.” Although their contributions to NASA were essential at the time, they faced discrimination in the work place in the early 1960s.

mathematical experience

2) The Man from the Future: The Visionary Life of John von Neumann by Ananyo Bhattacharya. This is a biography about John von Neumann, a child prodigy of mathematics, born in Budapest, Hungary, who would grow up to create the first ever digital computer and theorized the future existence of nanotechnology. Known to be smarter than Albert Einstein, he also contributed to logic/sets, quantum mechanics, games theory, nuclear strategy, artificial intelligence, and more! This book takes us on a tour of Von Neumann’s life and ideas and is an interesting read for anyone looking to learn more about this genius scientist from the early 1900’s.

mathematical experience

3) Alan Turing: The Enigma by Andrew Hodges. Another book that eventually became the movie you may know as, “The Imitation Game.” This book is the story of Alan Turing, a British mathematician that helped saved the Allies from Nazis in WWII with his universal machine, a foundation of the modern computer as we know it! Turing creates his machine and uses to crack enigma, the German code, to unlock their secrets and eventually win WWII. Interesting to find the differences between the book and the movie to see real and imaginary history, as the book has much more mathematically intense and technical material when compared to the story of the movie. Good read for those who want to learn more about Turing and the math behind his machine!

conceptual development

4) Logicomix: An Epic Search for Truth by Apostolos Doxiadis and Christos H. Papadimitriou, illustrated by Alecos Papadatos and Annie Di Donna. This is a graphic novel and biography about the philosopher Bertrand Russell as he searches for the logical foundations of mathematics and connects with other mathematicians. This book takes us on a journey back in time to Europe during WWI and shows Russell’s life including both his mental and professional struggles. With beautiful illustrations, this is an interesting read for anyone looking to learn more about Bertrand Russell.

conceptual development

Mathematics History for Kids

The list of top math books for kids is great for classrooms, libraries and who are we kidding, adults will also enjoy these books too! This a great way to learn more about math in a kid friendly and easy to understand way. Many of these books provide great illustrations that are perfect for visualizing stories as well as math.

1) A Quick History of Math: From Counting Caveman to Computers (Quick Histories) by Clive Gifford, illustrated by Michael Young. This book is for children aged 8-12 and is choc full of illustrations, that beautifully represent math visually for better understanding of the subject. Perfect at home or in the classroom as it makes the history of math fun for kids in grades 3-7. Be prepared to have fun reading and dive into the ancient history of mathematics including the invention of zero and the pythagorean theorem, while picking up fun math tips and tricks along the way!

mathematical experience

2) The Girl With a Mind for Math: The Story of Raye Monatague by Julia Finley Mosca, illustrated by Daniel Rieley. A story about a little girl who defies all odds to reach her dreams of becoming an engineer! Her aptitude for math and STEM is amazing, but her determination, fearlessness and humor is at the heart of this story of mathematics! This is an amazing and inspiring book with technical and biographical back matter in the back. Did I mention this book is based on the real life story of Raye Montague?! Montague was the first female program manager of ships in the United States Navy. Great for classroom or library book to have for discussion.

3) Whats the Point of Math? by DK. Written for children in grades 4 through 7, I honestly think this book could stretch even beyond that as it can appeal to younger kids and even adults. All ages should be able to enjoy a book that makes math simple and easy to understand while using illustrations. Learn how the origins of mathematical concepts such as sequences, pattern recognition, and trigonometry and see how these topics impact our lives in modern times. While reading this book, enjoy mathematical brain teasers, puzzles, games, math magic tricks, and fun facts along the way! Increase mathematical thought for your kids, for your students, or maybe just yourself!

Have you read any of the above books? What did you think? Is there a book on mathematics you think is missing from the list? Let me know in the comments below!

Also, don’t forget to follow us on social media links below. Thanks for stopping by and Happy calculating!

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Trig Identities

Hi everyone and welcome to MathSux! In today’s post, we are going to dip our toes into Trig Identities! There are a ton of trig identities out there but six trig functions that you’ll need to know for trig identity proofs, luckily most of them are related to the trigonometric functions you are probably already familiar with! Trig proofs involve working with trig functions we are already familiar with (sin θ, cos θ, tan θ), but breaking them down, inverting each trig function, and applying rules that are always true (otherwise known as trigonometric identities).

Proofs can feel a bit tricky at first, but with some practice they should start to feel more normal! The key is to always leave one side of the equation alone, while working with and manipulating the other side of the equation until it matches the other.

This is a great topic for getting more familiar with trig without actually having to work with any triangles. If you are familiar with the trig identities below you should feel good about answering these types of questions. Below you’ll also find a trig identity cheat sheet and a list of even more trigonometric identities that you may need! Also, don’t forget to check out the video and practice questions below to master trigonometric identities.  Happy calculating!

Trig Functions Identity Cheat Sheet:

Inverse Trigonometric Identities:

Inverse Trigonometric Identities, are the inverses of the same trig functions we all know and love (sin θ, cos θ, tan θ). When we take the inverse of sin θ, cos θ, tan θ, we end up getting these new trig functions: csc θ=1/sin θ, sec θ=1/cos θ, and cot θ=1/tan θ!  The trig functions and their inverses should all look familiar, so memorizing them should piece a piece of cake! Simple? Yes! These are a big key to solving many trig identity proofs.

Trig Identities

Trigonometric Ratio Identities:

Trigonometric Ratio Identities are great for breaking down tan θ and cot θ into tan θ=sin θ/cos θ and cot θ=cos θ/sin θ. Notice that tan and cot are the reciprocal of one another, all we need to do is remember the following:

Trig Identities

Trigonometric Pythagorean Identities:

Trigonometric Pythagorean Identities are based on none other than the pythagorean theorem in relation to trigonometry and the unit circle. The main equation to know for all of the listed Pythagorean Identities, is sin 2 θ+cos 2 θ=1. Knowing this one equation allows us to derive 8 more related pythagorean identity functions which will ultimately help us when breaking down trig functions in our trigonometric identity proofs. Below front and bolded is our main Pythagorean Identity:

Trig Identities

Now that we have everything we need to prove trig identities true, let’s apply our new knowledge and  take a look at an example below:

Example:

Step 1: First, we are going to focus on the left side of the equation only, trying to get it equal to the right side, leaving csc θ+cot θ unchanged.

Trig Identities

Step 2: Notice, we can expand the left side of the equation out, separating the sinθ in the denominator under each term of the numerator. This is based on basic rules for adding and subtracting fractions…nothing new or related to trig functions yet!

Trig Identities

Step 3: Now, looking at our trigonometric inverse and ratio identities, we can see that 1/sin θ can be re-written as the trig functions inverse identity, csc θ and that cos θ/sin θ can be re-written as trigonometric ratios identity as cot θ.  Let’s re-write those and fill them in below to get the correct answer and prove our trigonometric functions equation true:

Ready for more examples to try on your own!? Check out the practice problems below:

Practice Problems:

Prove each of the following true, then check your answer, with each worked through trig functions identity proof below.

Solutions:

Looking for more trigonometric identities? Check these out below for your reference! The trigonometric identities below are used for specific questions separate from the proof examples shown earlier in this post. But there will come a time when these will be needed so hold on to them for now and let me know if you’d like to see some examples!

Co-Function Identities:

Each of the Co-Function Identities below represents each basic trig function and their corresponding complementary angle, meaning that they each add to 90º (or in radians π/2).

trigonometric identities

Half Angle Identities:

trigonometric identities

Double Angle Identities:

trigonometric identities

Sum and Difference Identities:

Also known as ptolemy’s identities, the following are the sum and difference formulas for sine and cosine.

Still got questions? Have a favorite trigonometric identity? Don’t hesitate to email or comment below with any questions to clear things up! Happy calculating!

Also, don’t forget to follow us on social media to get the latest and greatest MathSux lessons, videos, questions, and more!

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Related Trigonometry Posts:

The Unit Circle

Basic Right Triangle Trigonometric Ratios (SOH CAH TOA)

4545 90 Special Triangles

30 60 90 Special Triangles

Graphing Trigonometric Functions

Transforming Trig Functions

Factoring Trigonometric Functions

Law of Cosines

Law of Sines

Circle Theorems

Greetings and welcome to MathSux! In today’s post we are going to go over everything you need to know about circles. We will start by going over the different parts of a circle including the centre of the circle, radius, chord, tangent, and secant. Then we will break down different formulas of a circle to know which include area, circumference, area of a sector, and several different circle theorems used to find missing angles and arcs.

Below is a cheat sheet that sums everything up in this post, but if you keep reading, we’ll get into more detail about how these theorems work with different examples and explanations. So keep scrolling, keep reading, stay positive, and happy calculating!

Free Downloadable Circle Theorem Cheat Sheet:

Parts of a Circle:

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There are so many different parts to a circle! Let’s take a closer look at each part below with the following definitions:

  • Diameter: A line that cuts a circle in half through its center.
  • Radius: A line that goes from the center point to the edge of the circle. (Otherwise known as half the length of the diameter).
  • Center: All points along the circle are equidistant from this point. 
  • Chord: A line segment that has endpoints along a circle but does not cross its center.
  • Tangent: A continuous line that only touches the outer part of the circle.
  • Secant: A line that cuts through a circle at two points.

Circle’s Circumference:

A circle’s Circumference, measures the distance of length around the entire circle. It is like we are taking a walk around the circle and measuring how far we go along the way until we’re back where we started.

Area of a Circle:

The Area of a circle allows us to find the “area” or total value found within outline of a circle.

Circle Theorems

Area of a Sector:

We already know how to find the area of a circle, but what about the area of a sector? What does a sector even mean? A Sector is a piece of a circle, kind of like a piece of pie. Check out the example here for a clearer picture.

Arcs, Angles, & Measures of a Circle:

The Arc of a circle is a piece of the circle’s circumference.  Think of an arc like the outline of a piece of the pie, yum!

What’s with the degrees and arc AB notation ? Is there a difference?

An arc is measured in degrees by its central angle value (called an arc’s measure).  An arc is also measured by length in units of measurement such as inches or centimeters. Let’s look at the difference between each below:

Measure: The measure of an arc is the degree size of its central angle. In the example below, we can see that the degree value of arc AB is 90º (hi-lighted in green)

Circle Theorem

Length: The length of an arc is the length of the circle’s circumference and can be measured in units such as feet, inches, centimeters, etc. (hi-lighted in pink)

Circle Theorem

Are there different types of arcs?

There are two main different types of arcs: a major arc and a minor arc. As you may guess, one is bigger and one is smaller.  Let’s look at an Example:

Major Arc: An arc with a measure value greater than 180º (greater than half the circle). In the Example below, we can see that the major arc can be represented by  arc AB (hi-lighted in green).

Minor Arc: An arc with a measure value less than 180º (less than half the circle). In the Example below, we can see that the minor arc can be represented by  arc AB (hi-lighted in pink).

circle theorem

How do we calculate an arcs Length and Measure?

Finding Arc Length: To find the length of an arc, we only need one formula!

Finding Arc Measure: Calculating an arc’s measure varies depending on the presence of secants, tangents, chords, and radii. In fact, there are seven different potential situations for finding arcs measure! Surprising, I know, but  let’s look at each type one at a time:

  1. Central Angle (Two radii): When two radii are drawn from the center point of a circle, they form a central angle. A central angle is equal to the length of the arc. In the Example below, we see that arc AB has a length of 90º and  has an arc measure of 90º.  They are equal!
Circle Theorems

For more on central angles, check out the post here for practice questions and the video below:

2. Inscribed Angle: When two chords come together to touch the outline of a circle, they create something called an inscribed angle. An inscribed angle is equal to half the value of the arc length.

Circle Theorems

For more on inscribed angles, check out the post and practice questions here, and video below!

3. Intersecting Chords: When two chords intersect, they create four arcs and two sets of vertical angles.  Each set of vertical angles is congruent.  To find the value of one vertical angle, add the two arc lengths together and divide by 2.

4. Tangent and Chord: When a tangent and chord connect, they create an angle that touches the outline of the circle. The angle formed is equal to half the arc length.

Circle Theorems

5. Two Tangents: When two tangents touch the outer edge of the circle, it creates an angle.  The angle is equal to the difference between the intercepted arc lengths divided by two.

6. Two Secants: When two secants intersect outside the circle, it creates one angle and two intercepted arc lengths.  The angle is equal to the difference between the intercepted arc lengths divided by two.

Circle Theorems

For more on Intersecting Secants, check out the video below and link here for practice questions!

7. Secant and Tangent: When an angle is formed by a secant and tangent it creates one angle and two intercepted arc lengths.  The angle is equal to the difference between the intercepted arc lengths divided by two.

Circle Theorems

For more Secant & Tangent review, check out the video below!

Circle Theorems:

Theorem 1: In a circle, when an angle is formed by a tangent and radius it creates an angle.  This happens always and every time!

Circle Theorems

Theorem 2: In a circles, inscribed angles that intercept the same arc, have equal angles. These types of overlapping arcs can also be known as “angles subtended by an arc.” In the example below, angle A and angle B are angles subtended by arc CD.

Circle Theorems

Theorem 3: An inscribed angles in a semi-circle is a right angle.

Circle Theorems

Theorem 4: When a quadrilateral is inscribed in a circle, opposite angles are supplementary (add to 180º). Notice below, opposite angles A and C are supplementary, adding to 180º.

Circle Theorems

*Fun Fact! A quadrilateral inscribed in a circle is a called a Cyclic Quadrilateral!

Theorem 5: In a circle, congruent central angles, also have congruent arcs (or vice versa).

Theorem 6: In a circle, congruent central angles, also have congruent chords (or vice versa).

Now that wraps up all we need to know about circles, yay! Although we are done reviewing what we need to know, we must prepare for the questions that apply our new circle knowledge.  One Example might go something like this:

Given circle R has arc BC=95º, two tangents AB and  AC and two radii RB and RC, find the following angles and arcs.

Circle Theorems

Before we just look at the solution, make sure to try this on your own!  Remember all the answers to finding the arcs and angles of Circle R are based on the circle theorems and seven different ways of finding angles that we just went over in this post.  So, go back if you need to, I know I needed to!

And now for that long-awaited Solution:

Circle Theorems

Explanation:

Still got questions?  No problem! Check out the videos above or comment below for any questions and follow for the latest Free math lessons, videos, and practice questions! Happy calculating! 🙂

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Tip! Also, don’t forget to check out the links found throughout this post for each related related lesson on central angles, inscribed angle theorems, intersecting secants theorem, and area of a sector, to dive even further into practice questions, videos, and more!

How to Study Math

How to Study Math

Greetings math friends! Today we are going to go over a question I get asked a lot, which is “How Do You Study Math?” How to study math, what a question! There is really only one way that I’ve ever known and that is to practice questions, over and over again. That is the basic advice I usually give because it is a quick and easy answer (and it’s true), but of course, there is more to the story! There are tips and tricks to practicing questions to master not just math, but any subject when preparing for a test. In this post, there is no holding back, and I’m going to unleash all my test prep secrets! Hopefully, this article will come in handy for when it is crunch time and hopefully, maybe, just possibly this study guide on studying can make math just a bit more fun (you never know)! Either way, I hope it helps when you need it most, good luck and happy calculating!

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Solving Math Problems:

Math is an active learning subject and the key to mastering any math topic is to solve similar math problems over and over again.

So in order to do this, of course, we are going to need practice questions. But what practice questions? How do I know which questions to look at?

That’s where we are going to gather and find every question we can on the topic som places to start might include:

1) Math Class Notes – Re-do and test yourself with questions from math class.

2) Homework Assignment Problems – Re-do homework problems, pay attention to those harder questions that you were unable to get the first time around. See how you do on them now, checking if you get the correct answer.

3) Quiz Questions – Were you already quizzed before the big test? How did you do? Review and learn if you got a wrong answer. Review and learn if you got the right answer!

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Now that we are equipped with all of our questions, let’s see what we should do next!

Cheat Sheet:

What am I forgetting? Do I have to memorize any formulas and what they mean? What does that notation represent again? Put it all on the cheat sheet!

Some math class teachers even allow you to bring a cheat sheet, if so you’re in luck! If not it is still a great tool that can be used for studying alone.

Here are some things you might want to include on your cheat sheet:

1) Math Formulas – We never want to say the word “memorize” in math class, but when there are going to be formulas on your math test, we know that’s exactly what we’ll need to do!

2) Math Vocabulary – Any new words you can’t seem to remember? Put them here with the definition for each new word.

3) Practice Questions – Sometimes we can forget how to do a certain type of question, if that happens, place the question fully solved here so we can remember how it’s done.

Don’t want to bother writing your own cheat sheet? Check out these free cheat sheets for Algebra, Geometry, Algebra 2/Trig, and Derivatives!

Now that we have done all we can on our own, let’s see I still have some math questions, what should I do now?

Study with Friends!

Forming a study group with friends is a great way to fill in any gaps you may have with math. What your friends know may be exactly what you’re missing when it comes to learning math and vice versa.

Study groups are also great for explaining topics you already know as this will increase your understanding even further! So even if you know more than your friends do in math class, you can teach and get even stronger in the topic, and become a math topic superhero!

Learning math can be fun with a group of friends as you can be more relaxed than you would be in a classroom and be more open to asking questions.

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Listen to Music:

Are you studying alone? A great way to study math concepts on your own while doing practice questions is to have some low relaxing music on in the background. This can help with concentration and allow some headspace for problem-solving and mathematical thinking. It can also help us relax and leave out any unnecessary chatter in our heads.

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Stay Positive!

A positive attitude to any math problem is so important for studying math. Make sure to know you can do it, you are capable of understanding the concepts and are capable of mastering the math course. If you have any questions you can always reach out to the resources available to you.

Explore Math Resources:

1) Math Class – Email or speak with a math teacher or tutor for any specific questions before your test.

2) Classmates / Friends – As mentioned before, use your circle of friends to fill any gaps in understanding!

3) MathSux / YouTube and other online math resources – Free and fast!

Math tests can be hard, but studying for them doesn’t have to be! Once you get in the flow of studying, the learning process of mathematics will come much easier. Do you have different study tips or a different method of studying math than the ones listed here? What works for you? Let me know in the comments and happy calculating!

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Shape Transformations in Math

Hi everyone and welcome to another week of MathSux! In today’s post, we are going to go over all the different types of shape transformations in math that we’ll come across in Geometry! Specifically, we’ll see how to translate, reflect, rotate, or dilate a shape, a line, or a point. There are also specific coordinate rules that apply to each type of transformation, but do not worry because each rule can also be easily derived (except for those tricky rotations, keep an eye out for those guys!). If you like art or drawing, this is a great topic where we’ll have to use our artistic eye and our imagination for finding the right answer. We’ll also take a look at where you might use and see transformations in your everyday life! Hope you are ready, take a look below and happy calculating! 🙂

What is a Transformation in Math?

Mathematical Transformations, include a wide range of “things.”  And by “things” I mean reflections, translations, rotations, and dilations; Each fall under the umbrella known as “transformations.”  Alone any one of these is not difficult to master but mix them together and add a test and a quiz or two and it can get confusing.  Even the words “transformation “and “translation” can get confusing to us humans, as they sound very similar. But these are two different things.  A translation is a type of transformation. Let’s break down each of our new words before our brains explode:

Transformations:  When we take a shape or line and we flip it, rotate it, slide it, or make it bigger or smaller.  Basically, when we have a shape or line and we mess around with it a bit, it is a transformation.  The shape or line in question is usually graphed on a coordinate plane. Transformations include:

(1) Translations (slide it)

(2) Reflections (flip it)

(3) Rotations (rotate it)

(4) Dilations (make it bigger or smaller)

Shape Transformation:

1) Translations – When we take a shape, line, or point and we move it up, down, left, or right.

2) Reflections – When a point, a line segment, or a shape is reflected over a line it creates a mirror image.

3) Rotations – When we take a point, line, or shape and rotate it clockwise or counterclockwise, usually by 90º,180º, 270º, -90º, -180º, or -270º.

4) Dilations – When we take a point, line, or shape and make it bigger or smaller, depending on the Scale Factor.

Rigid Transformations:

Before we dive into our first type of transformation, let’s first define and explore what it means when a transformation maintains Rigid Motion. When a line or shape is transformed and the length, area and angles of the line and/or shape are unaffected by the transformation, it is considered to have Rigid Motion. Rigid transformations include Translations, Reflections, and Rotations (but not Dilations).

Now that we know which types of transformations mainatin rigid motion, let’s explore each type of transformation in more detail!

Translations:

Translations: When we take a shape, line, or point and we move it up, down, left, or right. Remember that this type of transformation is a rigid transformation, meaning the line or shape is translated, the length, area and angles of the line and/or shape are unaffected by the transformation.

Shape Transformations in Math

In the translation example above, we go start at square ABCD and translate each coordinate of the original square ABCD 6 units to the right and 2 units up to get our new transformed image square A|B|C|D| .

Translations Formula:

P(x,y) -> P| (x+h, y+k)

where….

h=Horizontal Shift (add (+) when moving right, subtract (-) when moving left)

k= Vertical Shift (add (+) when moving up, subtract (-) when moving down)

Horizontal Translation:

When we translate a point, line, or shape left or right, it is undergoing a horizontal translation along the x-axis. Any type of left or right movement on a coordinate plane is a horizontal translation.

How does this affect the x-coordinate? If the shape is being translated to the right, then we are adding units to the x-coordinate, and if the shape is shifting left then we are subtracting units from the x-coordinate.

Vertical Translation:

When we translate a point, line, or shape up or down, it is undergoing a vertical translation along the y-axis. Any type of up and down movement on a coordinate plane is a vertical translation.

How does this affect the y-coordinate? If the shape is being translated up, then we are adding units to the y-coordinate, and if the shape is being shifted down then we subtract from the y-coordinate.

Even though a horizontal shift or a vertical shift can happen when we move a shape, line, or point, many translations have a combo of the two!

How do Coordinates Change after a Translation?

The truth is there is no one unique rule for translations, but numbers will always be added or subtracted from the x and/or y coordinate values.  If something is translated to the right, then we add units to the x-value. On the other had if something is translated to the left, we subtract units from the x-value.  The same can be said for moving a shape up, we then add units to the y-value, and if a shape is translated down, we subtract units from the y-value. This gives us the following translation formula below:

Shape Transformations in Math


If we look at our example, when we translate original square ABCD to square A|B|C|D|  we end up translating each coordinate of original square  ABCD 6 units to the right and 2 units up. What we are really doing when we translate is adding 6 units to each x-coordinate as well as adding 2 units to each y-coordinate of the original figure square ABCD.  Check it out below:

Shape Transformations in Math
Shape Transformations in Math

For more on translations, check out the video below and practice questions here.

Reflections:

Reflections on a coordinate plane are exactly what you think! When a point, a line segment, or a shape is reflected over a line it creates a mirror image.  Think the wings of a butterfly, a page being folded in half, or anywhere else where there is perfect symmetry. Check out how we solve the reflections example below one step at a time!

Shape Transformations in Math

Step 1: First, let’s draw in line x=-2.

Shape Transformations in Math

Step 2: Find the distance each point is from the line x=-2 and reflect it on the other side, measuring the same distance or mirror image of each point. First, let’s look at point C, notice it’s 1 unit away from the line x=-2, to reflect it we are going to count 1 unit (the same distance) to the left of the line x=-2 and label our new point, C|.

reflections

Step 3: Next we reflect point A in much the same way! Notice that point A is 2 units away on the left of line x=-2, we then measure 2 units to the right of our line and mark our new point, A|.

reflections

Step 4: Lastly, we reflect point B. This time, point B is 1 unit away on the right side of the line x=-2, we then measure 1 unit to the opposite side of our line and mark our new point, B|.

reflections

Step 5: Finally, we can now connect all of our new points, for our fully reflected triangle A|B|C|.

Shape Transformations in Math

If you’re looking for more on reflections, check out the videos below and the practice questions right here.

Rotations:

Rotations are a type of transformation in geometry where we take a point, line, or shape and rotate it clockwise or counterclockwise, usually by 90º,180º, 270º, -90º, -180º, or -270º.

A positive degree rotation runs counter clockwise and a negative degree rotation runs clockwise.  Let’s take a look at the difference in rotation types below and notice the different directions each rotation goes:

Shape Transformations in Math

How do we rotate a shape?

There are a couple of ways to do this take a look at our choices below:

  1. We can visualize the rotation or use tracing paper to map it out and rotate by hand.
  2. Use a protractor and measure out the needed rotation.
  3. Know the rotation rules mapped out below.  Yes, it’s memorizing but if you need more options check out numbers 1 and 2 above!

Rotation Rules:

Shape Transformations in Math

Where did these rules come from?

To derive our rotation rules, we can take a look at our first example, when we rotated triangle ABC 90º counterclockwise about the origin. If we compare our coordinate point for triangle ABC before and after the rotation we can see a pattern, check it out below:

Shape Transformations in Math

The rotation rules above only apply to those being rotated about the origin (the point (0,0)) on the coordinate plane.  But points, lines, and shapes can be rotates by any point (not just the origin)!  When that happens, we need to use our protractor and/or knowledge of rotations to help us find the answer. Let’s take a look at the Example below:

Shape Transformations in Math

Step 1: First, let’s look at our point of rotation, notice it is not the origin we rotating about but point k!  To understand where our triangle is in relation to point k, let’s draw an x and y axes starting at this point:

rotations

Step 2: Now let’s look at the coordinate point of our triangle, using our new axes that start at point k.

Step 2: Next, let’s take a look at our rule for rotating a coordinate -90º and apply it to our newly rotated triangles coordinates:

rotations

Step 3: Now let’s graph our newly found coordinate points for our new triangle .

rotations about a point

Step 4: Finally let’s connect all our new coordinates to form our solution:

rotations about a point

For more examples and practice questions, check out the video below and link here.

Dilations:

Dilations are a type of transformation in geometry where we take a point, line, or shape and make it bigger or smaller, depending on the Scale Factor.

We always multiply the value of the scale factor by the original shape’s length or coordinate point(s) to get the dilated image of the shape. A scale factor greater than one makes a shape bigger, and a scale factor less than one makes a shape smaller. Let’s take a look at how different values of scale factors affect the dilation below:

Scale Factor >1 Bigger

Scale Factor <1 Smaller

Scale Factor=2

 In the below diagram the original triangle ABC gets dilated by a scale factor of 2.  Notice that the triangle gets bigger, and that each length of the original triangle is multiplied by 2.

Shape Transformations in Math

Scale Factor=1/2

Here, the original triangle ABC gets dilated by a scale factor of 1/2.  Notice that the triangle gets smaller, and that each length of the original triangle is multiplied by 1/2 (or divided by 2).

Shape Transformations in Math


Properties of Dilations:

There are few things that happen when a shape and/or line undergoes a dilation.  Let’s take a look at each property of a dilation below:

1. Angle values remain the same.

2. Parallel and perpendicular lines remain the same.

3. Length, area, and perimeter do not remain the same.

*Notice Dilations are a non rigid transformation!

Now that we a bit more familiar with how dilations work, let’s look at some examples on the coordinate plane:

Shape Transformations in Math

Step 1: First, let’s look at two corresponding sides of our triangle and measure their length.

Dilations


Step 2:
Now, let’s look at the difference between the two lengths and ask ourselves, how did we go from 3 units to 1 unit?

Remember, we are always multiplying the scale factor by the original length values in order to dilate an image. Therefore, we know we must have multiplied the original length by 1/3 to get the new length of 1.

Dilations

Dilating about a Point other than the Origin

Dilations


Step 1:
First, let’s look at our point of dilation, notice it is not at the origin! In this question, we are dilating about point m!  To understand where our triangle is in relation to point m, let’s draw a new x and y axes originating from this point in blue below.

Dilations

Step 2: Now, let’s look at coordinate point K, in relation to our new axes.

Step 3: Let’s use the scale factor of 2 and the transformation rule for dilation, to find the value of its new coordinate point. Remember, in order to perform a dilation, we multiply each coordinate point by the scale factor.


Step 4:
Finally, let’s graph the dilated image of coordinate point K. Remember we are graphing the point (6,4) in relation to the x and y-axis that stems from point m.

Dilations

If you’re looking for more on dilations, check out the video below and practice questions right here!

Transformations in the Real World?

If you think that you’ll never see real world use of transformations, think again! When playing the lovable game of Tetris, we are rotating shapes to clear lines, transforming each shape as we go.

Besides playing Tetris, Transformations in math can be found within the game itself, within its code. Game developers will need to be familiar with coordinate rules for how to flip and rotate a shape within their code for Tetris or any other game out there!

Tetris GIF - Find & Share on GIPHY

You can also think of real-life objects to transform (as opposed to just the digital ones mentioned above). This can be anything from parking a car to building a house, to landing an airplane. Can you think of transformations you use in your everyday life? Let us know in the comments!

Driving Tiffany Haddish GIF by NETFLIX - Find & Share on GIPHY

Still got questions about math transformations? No problem! Don’t hesitate to comment with any questions below. Want more math transformations? Don’t forget to check out the videos and practice questions for each linked throughout this article. Thanks for stopping by and happy calculating! 🙂

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Geometry Constructions

Greetings math friends! In today’s post we are going to go over several geometric constructions you’ll need to know in order to pass Geometry! We’ll go over each kind of geometric construction one step at a time with compass and straightedge. Geometry constructions can be a lot of fun and a great part of math you may have never known about! Hope you are ready to get your math and artistic skills flowing, so have your compass and straightedge handy as we tackle the following Geometry Constructions.

Geometric Constructions:

Bisect a Line Segment

Perpendicular Line Through a Point

Angle Bisector

Steps of Construction of 45 degree Angle

Equilateral Triangle

Altitudes of a Triangle (Acute, Obtuse, Right)

Square Inscribed in a Circle

Bisect a Line Segment (Using a Compass & Straightedge):

A Perpendicular Bisector does cuts a line in half at its midpoint, creating two equal halves. This will creates four 90º angles about the line.

Geometry Constructions

How to Bisect a Line Segment Step by Step:

Step 1: First, we are going to measure out a little more than halfway across the line AB by using a compass.

Step 2: Next we are going to place the compass on point A and swing above and below line AB to make a half circle.

Step 3: Keeping the same distance on our compass, we are then going to place the point of the compass onto Point B and repeat the same step we did on point A, drawing a semi circle.

Step 4: Notice the intersections above and below line AB!? Now, we want to connect these two points by drawing a line with a ruler or straightedge.

Step 5: Yay! We now have a perpendicular bisector! This cuts line AB right at its midpoint, dividing line AB into two equal halves.  It also creates four 90º angles.

Check out the full video with explanation below and original post here.

Perpendicular Line Through a Point Construction:

A Perpendicular Line Through a Point is very similar to a perpendicular bisector, but this time instead of “slicing” a line segment right down the middle, we are creating a perpendicular line through any point on our line segment. A perpendicular line in this case will also create four 90º angles. Take a look at how it works below:

Geometry Constructions

Step 1: First, we are going to gather materials, for this construction we will need a compass, straightedge, and markers.

Step 2: Notice that we need to make a perpendicular line going through point B that is given on our line.

Step 3: Open up our compass to any distance (something preferably short though to fit around our point and on the line).

Step 4: Place the compass end-point on Point B, and draw a semi-circle around our point, making sure to intersect the given line.

Step 5: Open up the compass (any size) and take the point of the compass to the intersection of our semi-circle and given line.  Then swing our compass above the line.

Step 6: Keeping that same length of the compass, go to the other side of our point, where the given line and semi-circle connect.  Swing the compass above the line so it intersects with the arc we made in the previous step.

Step 7: Mark the point of intersection created by these two intersecting arcs we just made and draw a perpendicular line going through Point B!

Check out the full video with explanation below and original post here.

Angle Bisector Construction:

An Angle Bisector is a line that evenly cuts an angle into two equal halves, creating two equal angles. Angle Bisectors are great because they cut any and every angle in half every time! Take a look at the construction process below:

Geometry Constructions

Step 1: Place the point of your compass on the point of the angle.

Step 2: Draw an arc that intersects both lines that stem form the angle you want to bisect.

Step 3: Take the point of your compass to where the lines and arc intersect, then draw an arc towards the center of the angle.

Step 4: Now keeping the same distance on your compass, take the point of your compass and place it on the other point where both the line and arc intersect, and draw another arc towards the center of the angle.

Step 5: Notice we made an intersection!? Where these two arcs intersect, mark a point and using a straightedge, connect it to the center of the original angle. 

Step 6: We have officially bisected our angle into two equal 35º halves.

Check out the full video with explanation below and original post here.

Steps of Construction of 45 Degree Angle:

Time to construct a 45º angle! The key to getting this construction right is knowing that 45º is half of 90º. Let take a look at how this construction is done with compass and straightedge step by step below!

Geometry Constructions

Step 1: Using a straightedge, draw a straight line, labeling each point A and B.

Step 2: Using a compass, place the point of the compass on the edge of point A and draw a circle.

Step 3: Keeping the same length of the compass, take the point of the compass to the point where the circle and line AB intersect. Then swing compass and make a new arc on the circle.

Step 4: Keeping that same length of the compass, go to the new intersection we just made and mark another arc along the circle.

Step 5: Now, take a new length of the compass (any will do), and bring it to one of the intersections we made on the circle.  Then create a new arc above the circle by swinging the compass.

Step 6: Keep the same length of the compass and bring it to the other intersection we made on our circle.  Then create a new arc above the circle.

Step 7: Mark a point where these two lines intersect and using a straightedge, connect this intersection to point A. Notice this forms a 90º angle.

Step 8: Now to bisect our newly made 90º angle, we are going to focus on the pink hi-lighted points where the original circle intersects with line AB and our newly made line.

Step 9: Using a compass (any length), take the compass point to one of these hi-lighted points and make an arc.

Step 10: Keeping that same length of the compass, go to the other hi-lighted point and make another arc.

Step 11: Now with a straightedge, draw a line from point A to the new intersection of arcs we just made.

Step 12: Notice we split or 90º angle in half and now have two equal 45º angles?!

Check out the full video with explanation below and original post here.

Equilateral Triangle Construction:

Equilateral Triangle: A triangle with three equal sides.  Not an easy one to forget, the equilateral triangle is super easy to construct given the right tools (compass+ ruler). Take a look below:

how to construct an equilateral triangle
Geometry Constructions

1. Using a compass, measure out the distance of line segment  Screen Shot 2020-08-25 at 4.19.02 PM.

 2. With the compass on point A, draw an arc that has the same distance as Screen Shot 2020-08-25 at 4.19.02 PM.

 3. With the compass on point B, draw an arc that has the same distance as Screen Shot 2020-08-25 at 4.19.02 PM.

4. Notice where the arcs intersect? Using a ruler, connect points A and B to the new point of intersection. This will create two new equal sides of our triangle!

Check out the full video with explanation below and original post here.

Constructing Altitudes of a Triangle (Acute, Obtuse, Right):

An Altitude is a perpendicular line drawn from the vertex of a triangle to the opposite side, creating a 90º angle.

Geometry Constructions

Check out how to find the Altitudes of an acute, obtuse and right triangle in the video below and post here.

In the video above, we will look at how to find the altitude of an acute obtuse, and right triangle. We will also find something called the orthocenter which is explained below.

How to Find the Orthocenter of Triangle with a Compass:

The Orthocenter is a point where all three altitudes meet within a triangle.

How to Construct the Altitudes of a Triangle

In order to find the orthocenter using a compass, all we need to do is find the altitude of each vertex. The point at which they meet is the orthocenter. Check out the video above to see how this works step by step.

Square Inscribed in a Circle Construction:

Geometry Constructions

Step 1: Draw a circle using a compass.

Step 2: Using a ruler, draw a diameter across the length of the circle, going through its midpoint.

Step 3: Open up the compass across the circle. Then take the point of the compass to one end of the diameter and swing the compass above the circle, making a mark.

Step 4: Keeping that same length of the compass, go to the other side of the diameter and swing above the circle again making another mark until the two arcs intersect.

Step 5: Repeat steps 3 and 4, this time creating marks below the circle.

Step 6: Connect the point of intersection above and below the circle using a ruler. This creates a perpendicular bisector, cutting the diameter in half and forming 90º angles.

Step 7: Lastly, use a ruler to connect each corner point to one another creating a square.

Check out the full video with explanation below and original post here.

Best Geometry Tools!

Looking to get the best construction tools? Any compass and straight-edge will do the trick, but personally, I prefer to use my favorite mini math toolbox from Staedler. Stadler has a geometry math set that comes with a mini ruler, compass, protractor, and eraser in a nice travel-sized pack that is perfect for students on the go and for keeping everything organized….did I mention it’s only $7.99 on Amazon?! This is the same set I use for every construction video in this post. Check out the link below and let me know what you think!

Still got questions? Looking for more constructions that you don’t see here? No problem! Don’t hesitate to comment with any questions and comments below. Happy calculating! 🙂

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