How to Construct an Equilateral Triangle?: Geometry

Happy Wednesday math peeps! This post introduces constructions by showing us how to construct an equilateral triangle by using a compass and straightedge (or ruler). For anyone new to constructions in geometry, this is the perfect topic for art aficionados since there is more drawing here than there is actual math!

What is an Equilateral Triangle?

Equilateral Triangle: An equilateral triangle is 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+ straightedge). Take a look below:

equilateral triangle
equal sides
Equilateral Triangle: Triangle with all equal sides

Construct an Equilateral Triangle Example:

equilateral triangle
line segment ab

Check out the GIF below to see how to construct an equilateral triangle step by step using a compass and straightedge with pictures and explanations below!

Solution:

How to Construct an Equilateral Triangle b c

What’s Happening in this GIF? 

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

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

 3. With the compass now placed on point B, draw an arc that has the same distance as line segment 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 intersection point. This will create two new equal sides of our triangle!

5. We have now officially constructed a triangle with all equal side lengths!

Constructions and Related Posts:

Looking to construct more than just an equilateral triangle? Check out these related posts on geometry constructions!

Construct a Perpendicular Bisector

Perpendicular Line through a Point

Angle Bisector

Construct a 45º angle

Altitudes of a Triangle (Acute, Obtuse, Right)

Construct a Square inscribed in a Circle

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? No problem! Don’t hesitate to comment with any questions. Happy calculating! 🙂

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Looking to learn more about triangles? Check out this post on right triangle trigonometry here! And if you’re looking for even more geometry constructions, check out the link here!

Expanding Cubed Binomials: Algebra 2/Trig.

Greetings math friends! This post will go over expanding cubed binomials using two different methods to get the same answer. We’re so used to seeing squared binomials such as, Screen Shot 2020-08-19 at 11.29.14 AM.png, and expanding them without a second thought.  But what happens when our reliable squared binomials are now raised to the third power, such as,Screen Shot 2020-08-19 at 11.29.48 AM?  Luckily for us, there is a Rule we can use:

Screen Shot 2020-08-18 at 10.12.33 PM

But where did this rule come from?  And how can we so blindly trust it? In this post we will prove why the above rule works for expanding cubed binomials using 2 different methods:

Screen Shot 2020-08-19 at 11.31.13 AM

Why bother? Proving this rule will allow us to expand and simplify any cubic binomial given to us in the future! And since we are proving it 2 different ways, you can choose the method that best works for you.

Method #1: The Box Method

Screen Shot 2020-08-18 at 10.14.37 PM

Step 1: First, focus on the left side of the equation by expanding (a+b)3:

Expanding Cubed Binomials

Step 2: Now we are going to create our first box, multiplying (a+b)(a+b). Notice we put each term of (a+b) on either side of the box. Then multiplied each term where they meet.

Screen Shot 2020-08-18 at 10.15.50 PM

Step 3: Combine like terms ab and ab, then add each term together to get a2+2ab+b2.

Expanding Cubed Binomials

Step 4: Multiply (a2+2ab+b2)(a+b) making a bigger box to include each term.

Expanding Cubed Binomials

Step 5: Now combine like terms (2a2b and a2b) and (2ab2 and ab2), then add each term together and get our answer: a3+3a2b+3ab2+b3.

Expanding Cubed Binomials
Screen Shot 2020-08-18 at 10.21.05 PM.png

Method #2: The Distribution Method

Screen Shot 2020-08-18 at 10.17.54 PM.png

Let’s expand the cubed binomial using the distribution method step by step below:

Expanding Cubed Binomials
Screen Shot 2020-08-18 at 10.21.05 PM.png

Now that we’ve gone over 2 different methods of cubic binomial expansion, try the following practice questions on your own using your favorite method!

Practice Questions: Expand and simplify the following.

Screen Shot 2020-08-18 at 10.21.56 PM

Solutions:

Screen Shot 2020-08-18 at 10.22.19 PM.png

Still, got questions?  No problem! Check out the video above or comment below! Happy calculating! 🙂

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**Bonus: Test your skills with this Regents question on Binomial Cubic Expansion!

Recursive Rule

Welcome to Mathsux! This post is going to show you everything you need to know about how to use a Recursive Formula by looking at three different examples of a recursive rule. Check out the video below for more of an explanation and test your skills with the practice questions at the bottom of this page.  Please let me know if you have any questions in the comments section below and happy calculating! 🙂

What is a Recursive Formula or a Recursive Rule?

A Recursive Formula is a type of formula that forms a sequence based on the previous term value.  The recursive rules for each formula vary, but we are always given the first term and a formula to find the consecutive terms in the recursive sequence.

Recursive formula can be written as an arithmetic sequence (a sequence where the same number is either added or subtracted to each subsequent term to form a pattern) and recursive formulas can also be written as arithmetic sequences (a sequence where the same number is either multiplied or divided to each subsequent term to form a pattern). We’ll go over an example of each but both types of recursive rules are treated the same exact way!

What does all of this mean?  Check out the example below for a clearer picture.

Example #1: Arithmetic Recursive Sequence

recursive rules

Step 1: First, let’s decode what these formulas are saying.

recursive rules

Step 2: The first term, represented by a1, is and will always be given to us. In this case, our first term has the value a1=2 and represents the first term of our recursive sequence.

a1= First Term=2

Step 3: We then plug in the value of our first term, which is a1=2 into our formula an+4 to get 2+4=6. The number 6 now has the value of our second term in the recursive sequence.

a1= 2 First Term

a2= (2)+4=6 Second Term

Step 4: Now we are going to continue the pattern, plugging in the value of each previous term to find the next consecutive terms in our recursive sequence.

The pattern can be more easily seen below. Notice we are able to find the value of all 5 terms of the recursive sequence for the solution only given the first term and recursive formula at the beginning of our question.

algebra 2 recursive formula
recursive formula examples

Step 5: We found the recursive sequence we were looking for: 2, 6, 10, 14, 18. Since the question was originally only asking for the value of the fifth term we know our solution only needs to be the value of the fifth term which is 18.

Example #2: Geometric Recursive Sequence

Step 1: First, let’s decode what these formulas are saying.

algebra 2 recursive formula

Step 2: The first term, represented by a1, is and will always be given to us. In this case, our first term has the value a1=1 and represents the first term of our recursive sequence.

a1= First Term=1

Step 3: We then plug in the value of our first term, which is a1=1 into our formula 2an+1 to get 21+1=3. The number 3 now has the value of our second term in the recursive sequence.

a1= 1 First Term

a2= 2(1)+1=3 Second Term

Step 4: Now we are going to continue the pattern, plugging in the value of each previous term to find the next term in our recursive sequence.

The pattern can be more easily seen below. Notice we are able to find the value of all 3 terms of the recursive sequence for the solution only given the first term and recursive formula at the beginning of our question.

algebra 2 recursive formula
algebra 2 recursive formula

***Note this was written in a different notation but is solved in the exact same way! This recursive formula is a geometric sequence.

Step 5: We found the recursive sequence we were looking for: 1,3,9. Since the question was originally only asking for the value of the third term we know our solution only needs to be the value of the third term which is 9.

Example #3:

Step 1: First, let’s decode what these formulas are saying.

Step 2: The first term, represented by a1, is and will always be given to us. In this case, our first term has the value a1=4 and represents the first term of our recursive sequence.

a1= First Term=4

Step 3: We then plug in the value of our first term, which is a1=4 into our formula 3an-1-2 to get 3(2)-1=5. The number 5 now has the value of our second term in the recursive sequence.

a1= 4 First Term

a2= 3(2)-1=5 Second Term

Step 4: Now we are going to continue the pattern, plugging in the value of each previous term to find the next term in our recursive sequence.

The pattern can be more easily seen below. Notice we are able to find the value of all 3 terms of the recursive sequence for the solution only given the first term and recursive formula at the beginning of our question.

Step 5: We found the recursive sequence we were looking for: 4,10,28. Since the question was originally only asking for the value of the third term we know our solution only needs to be the value of the third term which is 28.

algebra 2 recursive formula

Think you are ready to solve a recursive equation on your own?! Try finding the specific term in each given recursive function below:

Practice Questions:

Solutions:

algebra 2 recursive formula

Related Posts:

Looking to learn more about sequences? You’ve come to the right place! Check out these sequence resources and posts below. Personally, I recommend looking at the arithmetic sequence or geometric sequence posts next!

Arithmetic Sequence

Geometric Sequence

Finite Arithmetic Series

Finite Geometric Series

Infinite Geometric Series

Golden Ratio in the Real World

Fibonacci Sequence

Still, got questions? No problem! Don’t hesitate to comment below or reach out via email. And if you would like to see more MathSux content, please help support us by following ad subscribing to one of our platforms. Thanks so much for stopping by and happy calculating!

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***Bonus! Want to test yourself with a similar NYS Regents question on Recursive Formulas?  Click here! And if you want to learn about even more sequences, check out the link here!

Reflections: Geometry

Greetings and welcome to Mathsux! Today we are going to go over reflections geometry, one of the many types of transformations that come up in geometry. And thankfully, reflection transformation is one of the easiest types to master, especially if you’re more of a visual learner/artistic type person. In this post, we will go over a reflection across the line x=-2, but if you want to see a reflection across the x axis, the y axis, and the line y=x, please check out the math video below! So let’s get to it! And if you’re new to transformations, check out translations here.

What are Reflections?

A Reflection on a coordinate plane is exactly what you think! A reflection is a type of transformation in geometry where we reflect a point, a line segment, or a shape over a line to create a mirror image of itself.  Think of the wings of a butterfly, a page being folded in half, or anywhere else where there is perfect symmetry, each of these are great examples of a reflection!

Reflection Rules:

reflections geometry

If you are completely new to reflections, you may want to check out the video above that goes into deriving reflection rules over the different axis. But for those of you more comfortable with reflection, check out the example below where we reflect the image of a triangle onto itself over the line x=-2.

Example:

Screen Shot 2020-08-04 at 5.19.40 PM

Step 1: First, let’s draw in line x=-2. Note that whenever we have x equal to a number, we end up drawing a vertical line at that point on the x axis, in this case at x=-2.

reflections

Step 2: Find the distance each point is from the line x=-2 and reflect it on the other side, measuring the same distance.

First, let’s look at point C, notice it’s 1 unit away from the line x=-2 on the right. To reflect point c, we are going to count 1 unit but this time to the left of the line x=-2 and label our new point, C|.

reflections

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

reflections

Step 4: Lastly, we want to reflect coordinate 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 direction of our line, x=-2, and mark our new point, B|.

reflections

Step 5: Now that we have all the newly reflected coordinate points of our triangle, finally, we can now connect them all, for our fully reflected image of right triangle A|B|C|.

Notice our newly reflected triangle is not just a mirror image of itself, but when the original figure is reflected it actually ends up overlapping onto itself!? How did this happen? That is because this our reflection line came right down the middle of our original image, triangle ABC. Shapes that reflect onto themselves are a bit tricky but not impossible, just remember to measure out the distance of each coordinate point and reflections should be a breeze!

Rigid Motion:

Reflections are a special type of transformation in geometry that maintains rigid motion, meaning when a point, line, or shape is reflected the angles, and line segments retain their value. For example, if we were to measure the area of both right triangles, before and after reflection, we would find the areas to remain unchanged. Meaning the area of triangle ABC is equal to the area of triangle A|B|C| . Another rigid transformation includes rotations and translations.

Looking to practice your new reflection skills? Try the reflection practice problems below, with solutions to each question, to truly master the topic! Happy calculating!

Practice Questions:

reflections

Solutions:

Still got questions?  No problem! Check out the video above or comment below! There is also a bonus video if you scroll all the way down at the end of this post for anyone who wants to see how to reflect a line over the line x=2. See how it differs and how it is similar to the example shown here. Happy calculating! 🙂

Don’t forget to follow MathSux on social media for the latest tips and tricks to ace math!

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And if you are looking for more Transformations Check out the related posts below and let me know if you have any questions? Or maybe you just want to share your favorite type of transformation? Either way, happy calculating!

Translations

Dilations

Rotation

Piecewise Functions: Algebra

Greetings, today’s post is for those in need of a piecewise functions review!  This will cover how to graph each part of that oh so intimidating piecewise functions.  There’s x’s, there are commas, there are inequalities, oh my! We’ll figure out what’s going on here and graph each part of the piecewise-function one step at a time.  Then check yourself with the practice questions at the end of this post. Happy calculating! 🙂

piecewise functions

What are Piece-Wise Functions?

Exactly what they sound like! A function that has multiple pieces or parts of a function.  Notice our function below has different pieces/parts to it.  There are different lines within, each with their own domain.

Now let’s look again at how to solve our example, solving step by step:

piecewise functions example
Screen Shot 2020-07-21 at 10.02.41 AM
piecewise functions

Translation: We are going to graph the line f(x)=x+1 for the domain where x > 0

To make sure all our x-values are greater than or equal to zero, we create a table plugging in x-values greater than or equal to zero into the first part of our function, x+1.  Then plot the coordinate points x and y on our graph.

Screen Shot 2020-07-21 at 10.04.33 AM
Screen Shot 2020-07-21 at 10.05.00 AM.png
Screen Shot 2020-07-21 at 10.06.46 AM

Translation: We are going to graph the line  f(x)=x-3 for the domain where x < 0.

To make sure all our x-values are less than zero, let’s create a table plugging in negative x-values values leading up to zero into the second part of our function, x-3.  Then plot the coordinate points x and y on our graph.

piecewise functions
Screen Shot 2020-07-21 at 10.07.57 AM

Ready to try the practice problems below on your own!?

Practice Questions:

Graph each piecewise function:

piecewise functions examples

Solutions:

piecewise functions examples
piecewise functions examples

Still got questions?  No problem! Check out the video above or comment below for any questions. Happy calculating! 🙂

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***Bonus! Want to test yourself with a similar NYS Regents question on piecewise functions?  Click here.

Intersecting Secants Theorem: Geometry

Ahoy! Today we’re going to cover the Intersecting Secants Theorem!  If you forgot what a secant is in the first place, don’t worry because all it is a line that goes through a circle.  Not so scary right? I was never scared of lines that go through circles before, no reason to start now.

If you have any questions about anything here, don’t hesitate to comment below and check out my video for more of an explanation. Stay positive math peeps and happy calculating! 🙂

Wait, what are Secants?

Screen Shot 2020-07-14 at 10.07.54 PM

Intersecting Secants Theorem: When secants intersect an amazing thing happens! Their line segments are in proportion, meaning we can use something called the Intersecting Secants Theorem to find missing line segments.  Check it out below: 

Intersecting Secants Theorem

Let’s now see how we can apply the intersecting Secants Theorem to find missing length.

Screen Shot 2020-07-14 at 10.45.29 PM.png

Step 1: First, let’s write our formula for Intersecting Secants.

Intersecting Secants Theorem

Step 2: Now fill in our formulas with the given values and simplify.

Intersecting Secants Theorem

Step 3: All we have to do now is solve for x! I use the product.sum method here, but choose the factoring method that best works for you!

Intersecting Secants Theorem

Step 4: Since we have to reject one of our answers, that leaves us with our one and only solution x=2.

Screen Shot 2020-07-14 at 10.14.41 PM.png

Ready to try the practice problems below on your own!?

Practice Questions: Find the value of the missing line segments x.

Intersecting Secants Theorem
Intersecting Secants Theorem

Solutions:

Screen Shot 2020-07-20 at 9.30.55 AM.png

Still got questions?  No problem! Check out the video above or comment below for any questions and follow for the latest MathSux posts. Happy calculating! 🙂

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To review a similar NYS Regents question check out this post here.

Circle Theorems & Formulas

Central Angle Theorem

Inscribed Angles & Intercepted Arcs

Area of a Sector

Circle Theorems

Absolute Value Equations: Algebra

Happy Wednesday math friends! Today, we’re going to go over how to solve absolute value equations.  Solving for absolute value equations supplies us with the magic of two potential answers since absolute value is measured by the distance from zero.  And if this sounds confusing, fear not, because everything is explained below!

Also, if you have any questions about anything here, don’t hesitate to comment. Happy calculating! 🙂

Absolute Value measures the “absolute value” or absolute distance from zero.  For example, the absolute value of 4 is 4 and the absolute value of -4 is also 4.  Take a look at the number line below for a clearer picture:

Absolute Value

Now let’s see how we can apply our knowledge of absolute value equations when there is a missing variable!Absolute Value Equations exampleScreen Shot 2020-07-08 at 2.03.46 PM.pngAbsolute Value EquationsScreen Shot 2020-07-08 at 2.04.26 PM.pngAbsolute Value Equations

Screen Shot 2020-07-08 at 2.05.17 PM.png

Absolute Value EquationsNow let’s look at a slightly different example:

Absolute Value Equations exampleScreen Shot 2020-07-08 at 2.07.59 PM

Absolute Value Equations

Screen Shot 2020-07-08 at 2.08.26 PM.png

Absolute Value Equations

Screen Shot 2020-07-08 at 2.09.33 PMAbsolute Value Equations Screen Shot 2020-07-08 at 2.10.39 PM.pngAbsolute Value Equations

Practice Questions: Given the following right triangles, find the missing lengths and side angles rounding to the nearest whole number.

Absolute Value Equations examples

Solutions:

Absolute Value Equations solutions

Still got questions?  No problem! Check out the video the same examples outlined above. Happy calculating! 🙂

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Also, if you’re looking for a review on combining like terms and the distributive property, check out this post here.

How to use SOH CAH TOA: Geometry

Greeting math friends and welcome to Mathsux! In this post, we are going to start with the very basics of trigonometry by going over how to find a missing angle and/or side length of right triangles while using the famous trigonometric function sine, cosine, or tangent, (aka how to use SOH CAH TOA).  Woo hoo! These are the basics of right triangle trigonometry, and provide the base for mastering so many more interesting things to come in trigonometry! So, let’s get to it!

SOH CAH TOA is an acronym that stands for the following trig functions and parts of a right triangle. We’ll explain more in this post!

SOH CAH TOA

Sin =
Opposite/Hypotenuse
Cosine=
Adjacent/Hypotenuse
Tangent=
Opposite/Adjacent
Note that SOH CAH TOA works on right triangles only!

Also, if you have any questions about anything here, don’t hesitate to comment below or watch the video below. Also, don’t forget to subscribe to MathSux for FREE math videos, lessons, and practice questions every week. Happy calculating! 🙂

What does SOH CAH TOA stand for?

A Trigonometric Ratio, more commonly known as Sine, Cosine, and Tangent, are trig ratios that naturally exist within a right triangle.  This means that the sides and angles of a right triangle are in proportion within itself.  It also means that if we are missing a side or an angle of a right triangle, based on what we’re given, we can figure out what the value of the sides or angles are, based on these ratios!

Let’s take a look at what Sine, Cosine, and Tangent are all about!

How to use SOH CAH TOA

Ready for your first right angled triangle example? Check it out below!

SOH CAH TOA Example #1:

Screen Shot 2020-07-04 at 5.04.02 PM

Now let’s see how we can apply trig ratios when there is a missing side or angle in a right triangle!

Step 1: First, let’s identify the different sides of our right triangle depending on which angle we are focusing on, which in this case is a 60º angle. Based on the locations of our angle, we can label each side as the hypotenuse, adjacent, or the opposite.

Notice below, that the opposite side labeled x, is labeled the “opposite” side because it is opposite to our given angle, 60º. The remaining side is considered the adjacent side of our triangle because it is directly next to our given angle, 60º.

How to use SOH CAH TOA

Step 2: Now, let’s write out SOH CAH TOA. Notice the only trig function that uses both the hypotenuse and the opposite is sine! Knowing to use the sine function, let’s fill in our formula using the hypotenuse = 5 and opposite = x in order to find the value for missing side length x.

In order to use the sin function correctly, we’re going to need to plug in our given angle, which is 60º, and then set up our proportion. sin(60º)=x/5. By using the sine function, our calculator, and a little bit of algebra we’ll be able to solve for the unknown side.

How to use SOH CAH TOA

Ready for another example?! Check out another SOH CAH TOA problem using right triangles below!

Right Triangle Trig Example #2:

Screen Shot 2020-07-04 at 5.05.01 PM.png

Step 1: First, let’s identify the different parts of the right triangle we are given (the hypotenuse, adjacent, and the opposite). Notice in this example, we are given the adjacent and hypotenuse and need to find the value of the unknown angle, θ.

Step 2: Next, let’s write out our acronym, SOH CAH TOA, to see which trig function can help us with our question! Notice the only trig function that uses both adjacent and hypotenuse is cosine. This is what we will use to solve for the unknown angle, θ.

We use cosine, by setting up our proportion, cos(θ)=adjacent/ hypotenuse, knowing we can then plug in 12 for our adjacent value, and 13 for our hypotenuse value.

How to use SOH CAH TOA

Think you’re ready to test out SOH CAH TOA on your own? Try the following Practice Questions on your own!

Practice Questions:

Given the following right triangles, find the missing lengths and side angles rounding to the nearest whole number.

Solutions:

Screen Shot 2020-07-04 at 5.06.37 PM.png

Still got questions?  No problem! Check out the video the same examples outlined above and 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 Trig Functions

Transforming Trig Functions

Factoring Trig Functions

Law of Cosines

Trig Identities

Law of Sines

How to Solve Log Equations: Algebra 2/Trig.

Welcome to Mathsux! Today, we’re going to go over how to solve log equations, yay! But before we get into finding x, though, we need to go over what log equations are and why we use them in the first place…..just in case you were curious!

Also, if you have any questions about anything here, don’t hesitate to comment below or shoot me an email.  Happy calculating! 🙂

What are Log Equations?

Logarithms are the inverses of exponential functions.  This means that when graphed, they are symmetrical along the line y=x.  Check it out below!

How to Solve Log Equations

When on the same set of axis, notice how the functions are symmetrical over the line y=x:

How to Solve Log Equations

We use logarithms to find the unknown values of exponents, such as the x value in the equation,.  This is a simple example, where we know the value of x is equal to 2,(Screen Shot 2020-06-24 at 9.32.30 PM.png). But what if it were to get more complicated?  That’s where logs come in!

How to Solve Log Equations?

Logarithms follow a swooping pattern that allows us to write it in exponential form, let’s take a look at some Examples below:

How to Solve Log Equations

But wait there’s more! Logs have a set of Rules that makes solving log equations a breeze!

How to Solve Log Equations

We can use these rules to help us algebraically solve logarithmic equations, let’s look at an example that applies the Product Rule.

Example:

Screen Shot 2020-06-24 at 9.36.08 PM.png
Screen Shot 2020-06-24 at 9.36.50 PM
Screen Shot 2020-06-24 at 9.46.07 PM.png
Screen Shot 2020-06-24 at 9.38.32 PM

Try the following practice questions on your own!

Practice Questions:

Screen Shot 2020-06-24 at 9.39.16 PM.png

Solutions:

Screen Shot 2020-06-24 at 9.40.37 PM

Still got questions?  No problem! Check out the video that goes over the same example outlined above.  And for more info. on logarithms check out this post that goes over a NYS Regent’s question here.  Subscribe below to get the latest FREE math videos, lessons, and practice questions from MathSux. Thanks for stopping by and happy calculating! 🙂

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****Check out this Bonus Video on How to Change Log Bases****

Perpendicular & Parallel Lines Through a Given Point: Geometry

Happy Wednesday math friends! Today we’re going to go over the difference between perpendicular and parallel lines, then we’ll use our knowledge of the equation of a line (y=mx+b) to see how to find perpendicular and parallel lines through a given point.  This is a common question that comes up on the NYS Geometry Regents and is something we should prepare for, so let’s go!

If you need any further explanation, don’t hesitate to check out the Youtube video below that goes into detail on how to find perpendicular and parallel lines through a given point one step at a time. Happy calculating! 🙂

Perpendicular Lines:

Perpendicular & Parallel Lines Through a Given Point

Perpendicular Lines: Lines that intersect to create a 90-degree angle and can look something like the graph below.  Their slopes are negative reciprocals of each other which means they are flipped and negated. See below for example!

Example: Find an equation of a line that passes through the point (1,3) and is perpendicular to line y=2x+1 .

Screen Shot 2020-06-10 at 10.28.20 AM
Perpendicular & Parallel Lines Through a Given Point
Perpendicular & Parallel Lines Through a Given Point
Screen Shot 2020-06-10 at 10.29.06 AM

Parallel Lines:

Parallel lines are lines that go in the same direction and have the same slope (but have different y-intercepts). Check out the example below!

Perpendicular & Parallel Lines Through a Given Point

Example: Find an equation of a line that goes through the point (-5,1) and is parallel to line y=4x+2.

Screen Shot 2020-06-10 at 10.34.46 AM
Screen Shot 2020-06-10 at 10.35.23 AM

Try the following practice questions on your own!

Practice Questions:

1) Find an equation of a line that passes through the point (2,5) and is perpendicular to line y=2x+1.

 2) Find an equation of a line that goes through the point (-2,4) and is perpendicular to lineScreen Shot 2020-06-10 at 11.24.06 AM

 3)  Find an equation of a line that goes through the point (1,6) and is parallel to line y=3x+2.

4)  Find an equation of a line that goes through the point (-2,-2)  and is parallel to line y=2x+1.

Solutions:

Screen Shot 2020-06-10 at 11.22.05 AM

Need more of an explanation? Check out the video that goes over these types of questions up on Youtube (video at top of post) and let me know if you have still any questions.

Happy Calculating! 🙂

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Looking for more on Perpendicular and parallel lines? Check out this Regents question on perpendicular lines here!