Academic Archives - Page 2 of 8

How to Construct a Perpendicular Line through a Point on the Line

Greetings math peeps and welcome to another week of MathSux! In this post, we will learn how to construct a perpendicular line through a point on the line step by step. In the past, we learned how to bisect a line by constructing a perpendicular bisector right down the middle of a line segment, but in this case, we will learn how to create a perpendicular line through a given point on the line (which is not always in the middle). As always, please follow along with the GIF and step-by-step tutorial below or check out the video. Thanks for stopping by and happy calculating! 🙂

What are Perpendicular Lines?

Perpendicular lines are lines that intersect to create four 90º angles (or right angles) about the two line segments. In the example below, line l is perpendicular to line segment AB, which forms a right angle.

Segment Bisector — Line l is perpendicular to line segment AB

Note! When we construct a Perpendicular Bisector, the line we create forms a 90-degree angle and splits the line segment in half. In the construction below, however, we are creating a perpendicular line through a point already on the line segment. Note that the point given to us, will not always be splitting the line into two equal halves the way a segment bisector does. See for yourself below!

How to Construct a Perpendicular Line through a Point on the Line?:

What is happening in this GIF?

Step 1: First, notice we are given line segment AC with point B, not in the middle, but along our line. We are going to need a compass and a straightedge or ruler to complete our construction.

Step 2: Our goal is to make a perpendicular line going through point B that is given on our line segment AC.

Step 3: First, let’s open up our compass to any distance (something preferably short enough to fit around our point and on line segment AC).

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 segment.

Step 5: Next, open up the compass at any size and take the point of the compass to the intersection of our semi-circle and given line segment. Then swing our compass above line segment AC.

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: Now we can mark the point of intersection created by these two intersecting arcs we just made and draw a perpendicular line using a straight edge going through Point B and we have created our perpendicular line!

Perpendicular Bisector Theorem:

The Perpendicular Bisector Theorem explains that any point along the perpendicular bisector line we just create is equidistant to each end point of the original line segment (in this case line segment AB).

Therefore, if we were to draw points C,D, and E along the perpendicular bisector, then draw imaginary lines stemming from these points to each end point, we’d get something like the image below:

AC = CB

AD = DB

AE = EB

Constructions and Related Posts:

Looking to construct more than just a perpendicular bisector? Check out these related posts and step-by-step tutorials on geometry constructions below!

Construct an Equilateral Triangle

Bisect a Line Segment with Segment Bisector

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 below or check out the video above. Thanks for stopping by and happy calculating! 🙂

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Want to see how to construct a square inscribed in a circle? Or maybe you want to construct an equilateral triangle? Click on each link to view each construction! And if you’re looking for even more geometry constructions, check out the link here!

Origami and Volume of a Box and Square Base Pyramid

Greetings and happy summer math peeps! In honor of the warm weather and lack of school, I thought we’d have a bit of fun with origami and volume! In this post, we will find the volume of a box and the volume of a square base pyramid. We will also be creating each shape by using origami and following along with the video below. For anyone who wants to follow along with paper folding tutorial, please note that we will need one piece of printer paper that is 8.5″ x 11″and one piece of square origami paper that is 8″ x 8″. If you’re interested in more math and art projects check out this link here. Stay cool and happy calculating! 🙂

Volume of Box (or Rectangular Prism):

To get the volume of our origami box (video tutorial above), we are going to multiply the length times the width times the height. All the values and units of measurement were found by measuring the box we made in inches in the video above with 8.5 x 11 inch computer paper.

Volume of Square Base Pyramid:

Below is a diagram of the square base pyramid we created via paper folding (watch video tutorial above to follow along!). Please note that if you used a different sized paper (other than 8 X 8 inches), you will get a different value for measurements and for volume.

For step by step instruction, don’t forget to check out the video above to see how to paper fold a box and square base pyramid. I hope this post made math suck just a little bit less and finding volume a bit more fun. Still got questions or want to learn more about Math+ Art? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! 🙂

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For more Math + Art, check out this post on Perspective Drawing here. And for more Volume + Origami, check out this post on how to find the volume of a cube using origami here and below! If you’re looking for more math and crafts, learn how to make a Mobius Band here!

The Original Spirograph: Math + Art

Happy Summer everyone! Now that school is out, I thought we could have a bit of fun with Math and Art! In this post, we will go over how to make a the original spirograph (by hand) step by step using a compass and straight edge. Follow along with the video below or check out the tutorial in pictures in this post. Hope everyone is off to a great summer. Happy calculating! 🙂

What is a Spirograph?

The childhood toy we all know and love was invented by Denys Fisher, a British Engineer in the 1960’s.But the method of creating Spirograph patterns was invented way earlier by engineers and mathematicians in the 1800’s.

The Original Spirograph (by hand):

Step 1: Gather materials, for this drawing, we will need a compass and straight edge.

Step 2: Using our compass, we are going to open it to 7 cm and draw a circle.

Step 3: Next, we are going to open the compass to 1cm, making marks all around the circle, keeping that same distance on the compass.

Step 4: Draw a line connecting two points together (any two points some distance apart will do).

Step 5: Now, we are going to move the straight edge forward by one point each and connect the two points with another line.

Step 6: Continue this pattern of moving the ruler forward by one point and connecting them together all the way around.

Step 7: We have completed our Spirograph drawing! Try different sized circles, points around the circle, colors, and points of connections to create different types of patterns and have fun! 🙂

Spirograph Deluxe Art Set:

Want to try the one and only toy spirograph on your own!? Check out this Deluxe Spirograph set that brings mathematics and art together! Let your artistic creativity run free by experimenting with different-sized spirograph tools and colorful pens! Great for kids or math nerd adults, and easily available at Amazon for $23.99. Let me know what you think if you end up getting a spirograph set or if you already have one!

Still got questions or want to learn more about Math+ Art? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! 🙂

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For more Math + Art, check out this post on Perspective Drawing here. And for another math + art project, check out this post on Mobius Bands!

Dilations: Scale Factor & Points Other than Origin

Hi there and welcome to MathSux! Today we are going to break down dilations; what they are, how to find the scale factor, and how to dilate about a point other than the origin. Dilations are a type of transformation that are a bit different when compared to other types of transformations out there (translations, rotations, reflections). Once a shape is dilated, the length, area, and perimeter of the shape change, keep on reading to see how! And if you’re looking for more transformations, check out these posts on reflections and rotations. Thanks so much for stopping by and happy calculating! 🙂

What are 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.

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).

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.

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

Example #1: Finding the Scale Factor

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

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.

When it comes to dilations, there are different types of questions we may be faced with. In the last question, the triangle dilated was done so about the origin, but this won’t always be the case. Let’s see how to dilate a point about a point other than the origin with this next example.

Example #2: Dilating about a Point other than the Origin

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.

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.

Check out these dilation questions below!

Practice Questions:

1) Plot the image of Point Z under a dilation about the origin by a scale factor of 2.

2) Triangle DEF is the image of triangle ABC after a dilation about the origin. What is the scale factor of the dilation?

3) Point L is dilated by a scale factor of 2 about point r. Draw the dilated image of point L.

4) Line DE is the dilated image of line AB. What is the scale factor of the dilation?

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! 🙂

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Looking for more Transformations? Check out the related posts below!

Translations

Rotations

Reflections

Algebra 2 Cheat Sheet & Review

Ahoy and welcome math friends! For the latest installment, here is the Algebra 2 Cheat Sheet & Review made just for you to prepare for finals. On this page, you’ll also find links to the come math friends! For the latest installment, here is the Algebra 2 lesson playlist, the NYS Algebra 2 Common Core Regent’s Playlist, and the library of Geometry blog posts. Hope you find these resources helpful as the end of the school year approaches. Good luck on finals and happy calculating! 🙂

Algebra 2 Cheat Sheet:

Download and print the below .pdf for a quick and easy guide of everything you need to know for finals; From formulas to graphs, it’s on here.

Algebra-2Trig.-Cheat-Sheet-1 Download

Algebra 2 Playlist:

Looking for a more detailed review? Check out the Youtube playlist for Algebra 2 below. It includes every MathSux video related to Algebra 2 and will be sure to help you ace the test!

Algebra 2 Common Core Regents Review:

This playlist is made especially for New York State dwellers as it goes over each and every question of the NYS Common Core Regents. Perfect if you are stuck on that one question! You will surely find the answer here.

Algebra 2 Blog Posts:

For anyone in search of blog posts and practice questions, check out MathSux’s entire Algebra 2 library organized by topic here.

Still got questions? No problem! Don’t hesitate to comment with any questions below. Also, if you find you need some motivation, check out my 6 tips and tricks for studying math here! Thanks for stopping by and happy calculating! 🙂

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Get everything you need to know with this Algebra 2 Cheat Sheet and Review! Download and print the pdf for reviewing Algebra 2 or check out the video playlists for a more in-depth review of each topic. If you are living in NYS, you also might want to check out the NYS Regents Common Core Video as needed!

Geometry Cheat Sheet & Review

Greeting math peeps! As promised here is the Geometry Cheat Sheet and Review made just for you to prepare for finals. On this page, you’ll also find links to the Geometry lesson playlist, the NYS Geometry Common Core Regent’s Playlist, and the library of Geometry blog posts. Hope you find these resources helpful as the end of the school year approaches. Good luck on finals and happy calculating! 🙂

Geometry Cheat Sheet:

Download and print the below .pdf for a quick and easy guide of everything you need to know for finals; From formulas to shapes, it’s on here.

Geometry-Cheat-Sheet-2 Download

Geometry Review Playlist:

Looking for a more detailed review? Check out the Youtube playlist for Geometry below. It includes every MathSux video related to Geometry and will be sure to help you ace the test!

Geometry Common Core Regents Review:

Geometry Math Lessons for Review:

For anyone in search of blog posts and practice questions, check out MathSux’s entire Geometry library organized by topic here.

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Get everything you need to know with this Geometry Cheat Sheet and Review! Download and print the pdf for reviewing Geometry or check out the video playlists for a more in-depth review of each topic. If you are living in NYS, you also might want to check out the NYS Regents Common Core Video as needed!

Algebra Cheat Sheet & Review

It’s that time of year again, summer is coming, the vacation vibes are calling, but so, unfortunately, are the test cramming and non-stop class reviewing that is coming our way. Nothing like going over topics mentioned at the beginning of the school year to bring us down. How is one supposed to remember everything? Fear not, because I have made a special cheat sheet and review for Algebra, (with Geometry and Algebra 2/Trig. soon to be on the way). I hope you’re staying safe, cool, and calm as the end of the year approaches. Good luck on finals and tests and happy calculating! 🙂

Algebra Cheat Sheet:

Download and print the below .pdf for a quick and easy guide of everything you need to know for finals; From formulas to parabolas, it’s on here.

Algebra-Cheat-Sheet-1 Download

Algebra Playlist:

Looking for a more detailed review? Check out the Youtube playlist for Algebra below. It includes every MathSux video related to Algebra and will be sure to help you ace the test!

Algebra Common Core Regents Review:

Algebra Blog Posts:

For anyone in search of blog posts and practice questions, check out MathSux’s entire Algebra library organized by topic here.

Still got questions? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! 🙂

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Get everything you need to know with this Algebra Cheat Sheet and Review! Download and print the pdf for reviewing Algebra or check out the video playlists for a more in-depth review of each topic. If you are living in NYS, you also might want to check out the NYS Regents Common Core Video as needed!

Similar Triangles: AA, SSS, & SAS

Happy Wednesday math peeps! In today’s post, we are going to go over Proving Similar Triangles, by going over:

1) What it means when two triangles are similar?

2) How to prove two triangles similar?

3) How to find missing side lengths given triangles are similar?

For even more practice, don’t forget to check out the video and practice problems below. Happy calculating! 🙂

What are Similar Triangles?

When two triangles have congruent angles and proportionate sides, they are similar. This means they can be different in size (smaller or larger) but as long as they have the same angles and the sides are in proportion, they are similar! We use the “~” to denote similarity.

In the Example below, triangle ABC is similar to triangle DEF:

How can we Prove Triangles Similar?

There are three ways to prove similarity between two triangles, let’s take a look at each method below:

Angle-Angle (AA): When two different sized triangles have two angles that are congruent, the triangles are similar. Notice in the example below, if we have the value of two angles in a triangle, we can always find the third missing value which will also be equal.

Side-Side-Side (SSS): When two different sized triangles have three corresponding sides in proportion to each other, the triangles are similar.

Side-Angle- Aside (SAS): When two different sized triangles have two corresponding sides in proportion to each other and a pair of congruent angles between each proportional side, the triangles are similar.

Let’s look at how to apply the above rules with the following Example:

Step 1: Since, we know the triangles ABC and DEF are similar, we know that their corresponding sides must be in proportion! Therefore, we can set up a proportion and find the missing value of length EF by cross multiplying and solving for x.

Practice Questions:

1) Are the following triangles similar? If so, how? Explain.

2) Are the following triangles similar? If so, how? Explain.

3) Given triangle ABC is similar to triangle DEF, find the side of missing length AB.

4) Given triangle ABC is similar to triangle PQR, find the side of missing length AC.

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! 🙂

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Check out more posts on Geometry here!

And if you’re looking for more lessons on triangles, check out these related posts below:

Congruent Triangles

Similar Triangles

45 45 90 special triangles

30 60 90 special triangles

Construct the Altitude of a Triangle

Legs of a Right Triangle (when an altitude is drawn)

What is the Discriminant?

Hi everyone and welcome to MathSux! In this post, we are going to answer the question, what is the discriminant? Before going any further, if you need a review on what the quadratic equation or imaginary numbers are, check out each related link! Also, don’t forget to check out the video and practice questions below. Happy calculating! 🙂

What is the Discriminant?

The discriminant is a formula we can use that tells us more about a quadratic equation including:

The number of solutions a quadratic equation has.
The “nature” of the roots of the solution (rational/irrational or real/imaginary).

Discriminant Formula:

The discriminant formula may look familiar! It is part of the quadratic formula and we have seen it before, using the very same coefficients a, b, and c from the quadratic equation.

How does it Work?

When we find the value of the discriminant of any quadratic equation, it will give us a value that tells us how many solutions (or roots) a quadratic equation has. Remember when we say “roots” what we really mean are the x-value(s) of the quadratic equation that hit the x-axis. This value will also tell us if the solutions to the quadratic equation are rational/irrational or real/imaginary. Take a look at how it all breaks down below:

Now that we are familiar with the rules, let’s take a look at an Example:

Step 1: First let’s write out our quadratic equation and identify the coefficients a, b, and c so they are ready to be plugged into our discriminant formula.

Step 2: Now let’s write out and fill in our formula using the coefficients and solve.

Step 3: Now let’s analyze our answer! Since, we got a discriminant value of 36, notice that it is a positive perfect square! If we look back at our discriminant table, this tells us that our quadratic equation is going to have 2 real and rational solutions.

Practice Questions:

Find the discriminant, number of solutions and nature of the roots of the following quadratic equations:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions below or check out the video above. Thanks for stopping by and happy calculating! 🙂

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Looking for more on Quadratic Equations and Functions? Check out the following Related posts!

Factoring Review

Factor by Grouping

Completing the Square

4 Ways to Factor a Trinomial

Is it a Function?

Imaginary and Complex Numbers

Focus and Directrix of a Parabola

Quadratic Equations with 2 Imaginary Solutions

Finite Geometric Series

Happy Wednesday math friends! In today’s post, we are going to go over what the phrase, “finding the sum of a finite geometric series” means and then use the finite geometric series formula to solve an example one step at a time! A finite geometric series happens when we add together a finite amount of terms from a geometric sequence together. We will go into more detail below and as always if you have any questions be sure to check out the video and comment with any questions below.

*If you need a refresher on geometric sequences (otherwise known as a geometric progression) before tackling these types of questions, don’t hesitate to check out this post! And, if you’re looking for an infinite geometric series, as opposed to a finite one, also be sure to check that out.

What does it mean to find the “Sum of the Finite Geometric Sequence”?

We already know what a geometric sequence is: a sequence of numbers that form a pattern when the same number is multiplied or divided to each subsequent term. In the example below, we can see that each number in the sequence is being multiplied by the common ratio, the number 2.

Geometric Sequence Example:

But what happens if we wanted to sum the first 20 terms of our geometric sequence together? Adding 2+4+8+16+……a₂₀ . Notice I used the notation a₂₀ to represent our unknown 20th term of the finite sequence.

Example:

How would we calculate that? Instead of finding the first 20 terms of our sequence and adding them all together, thankfully there is a better way, which is where our Finite Geometric Series formula comes in handy!

Why is it called “finite”? Adding the first 20 terms of our geometric sequence are considered to be “finite” because the first 20 terms have a definite ending as opposed to a sequence that is infinite and goes on forever. Adding together an infinite geometric series comes with a different formula.

Finite Geometric Series Formula:

a₁=The first term of our sequence. In this case, we can see that the first term will be the number 2 in the example above. Therefore, we can say a₁=2.

r= The common ratio is the number that is multiplied or divided by each consecutive term within the sequence. In the example above, each number is multiplied by 2, therefore we can say, r=2.

n= The total number of terms we are trying to sum together. In the example mentioned above, we are trying to sum 20 terms in total, so in this case n=20.

Now that we have this finite geometric series equation to work with, let’s take another look at our question and apply our finite geometric series formula to answer the solution:

Finite Geometric Series Example:

Step 1: First let’s write out the finite geometric series formula and identify what each part represents/what numbers need to be filled in.

a₁=2 The first term of our sequence. a₁=2.

r= 2 The common ratio is the number that is multiplied or divided by each consecutive term within the sequence.

n= 20 The total number of terms we are trying to sum together.

Step 2: Now let’s plug in our numbers into the finite geometric series formula and calculate and solve with the given values.

We have found our solution! Remember the number here, 2,097,150 represents the sum of the first 20 numbers of the geometric sequence given to us 2+4+8+16+……a₂₀=2,097,150.

Summation Notation – Finite Geometric Sequence

You may come across the finite geometric series formula in a different format, known as summation notation or sigma notation. Writing in summation notation, we are actually writing the sum of the geometric sequence in a different way, but will still come to the same solution as we did using the formula. Check it out below:

Now that we know what each part of this summation notation looks like, let’s actually identify what each part of this equation means:

Now let us go back and try to solve our original question, finding the sum of the first 20 terms of the sequence 1, 4,8, 16, …. all the way to thew 20th term:

Notice we get the same exact solution as we did in the previous example, mission accomplished!

Note! Does the above summation notation totally freak you out? Fear not! Learn more about how summation notation works here,what it means, and don’t be intimidated by these math symbols anymore!

Think you are ready to try questions on your own? Check out similar practice questions and find the sum of each finite series below!

Practice Questions:

1) Find the sum of the first 15 terms of the following geometric sequence:

4, 12, 36, 108, ….

2) Find the finite sum of the first 12 terms of the following sequence and round to the nearest tenth:

128, 64, 32, 16, ….

3) Find the sum of the first 18 successive term of the following geometric sequence and round to the nearest tenth:

400, 100, 25, 6.25, ….

4) Find the sum of the first 12 consecutive terms of the following geometric sequence:

3, 6, 12, 24, ….

Solutions:

1) 28,697,812

2) 255.9

3) 533.3

4) 12,285

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 finite geometric sequence or the geometric infinite series posts next!

Geometric Sequence

Recursive Formula

Arithmetic Sequence

Finite Arithmetic Series

Infinite Geometric Series

Golden Ratio in the Real World

Fibonacci Sequence

Still, got questions? No problem! Don’t hesitate to comment with any questions below or check out the video above. Thanks so much for stopping by and happy calculating! 🙂

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