Angle Bisector Definition & Example

Hi everyone and welcome to another fabulous week of MathSux! I bring to you the first construction of the back to school season! In this post, we are going to go over the angle bisector definition and example. First, we will define what an angle bisector is, then we’ll take our handy dandy compass and straight edge to construct an angle bisector that will bisect an angle for any size! Check out the video and GIF below for more and happy calculating! ๐Ÿ™‚

What is an Angle Bisector?

A line that evenly cuts an angle into two equal halves, creating two equal angles.

Angle Bisector Example:

Angle Bisector Definition & Example

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 straight edge, connect it to the center of the original angle.ย 

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

*Please note that the above example bisects a 70ยบ angle, but this construction method will work for an angle of any size!๐Ÿ™‚

What do you think of the above angle bisector definition & example? Do you use a different method for construction? Let me know in the comments below! ๐Ÿ™‚

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Looking for more constructions? Check out how to construct a square inscribed in a circle and an equilateral triangle by clicking on their respective links!

AnAngle Bisector Definition & Example

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 construct a perpendicular bisector right down the middle of the line, 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). Following along with the GIF or check out the vide below. Thanks for stopping by and happy calculating! ๐Ÿ™‚

What are Perpendicular Lines ?

  • Lines that intersect to create four 90ยบ angles about the two lines.
How to Construct a Perpendicular Line through a Point on the Line

What is happening in this GIF?

Step 1: First, we are going to gather materials, for this construction we will need a compass, straight edge, 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!

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! ๐Ÿ™‚

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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!

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! ๐Ÿ™‚

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):

The Original Spirograph

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

The Original Spirograph

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

The Original Spirograph

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.

The Original Spirograph

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

The Original Spirograph

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

The Original Spirograph

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! ๐Ÿ™‚

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! ๐Ÿ™‚

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For more Math + Art, check out this post on Perspective Drawing here.

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! ๐Ÿ™‚

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 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:

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.

Geometry Blog Posts:

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

Geometry Cheat Sheet & Review

Still got questions? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! ๐Ÿ™‚

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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!

Proving Similar Triangles: AA, SSS, & SAS

Happy Wednesday math peeps! TIn 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! ๐Ÿ™‚

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:

Similar Triangles

How can we Prove Triangles Similar?

There are three ways to prove similarity between two triangles, letโ€™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.

Similar Triangles

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

Similar Triangles

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. 

Similar Triangles

Letโ€™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.

Similar Triangles

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! ๐Ÿ™‚

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Check out more posts on Geometry here!

Math + Art: Math Behind Perspective Drawing

Greetings! We are going to do something a little different today and explore Math + Art: Math Behind Perspective Drawing. For all the artists out there who tend to not generally gravitate towards math, this post is for you! There are many ways math can be connected to art, and in this post we will explore the role parallel and perpendicular lines play when it comes to drawing 3-D shapes. And for those who want to learn even more, don’t forget to check out the video below to see how 2-point perspective applies geometry and angles to create 3-D shapes.

What is Perspective Drawing?

Perspective drawing is an art technique that allows us to draw real life objects in 3-D on a flat piece of paper. Notice in the example below that buildings, trees, and power lines get smaller and smaller as we look into the distance just as they would in real life.

Math Behind Perspective Drawing

What are the Basics of Perspective Drawing?

Math Behind Perspective Drawing

There are two main things we need to know about perspective drawing.

1- Horizon Line: A horizontal line that goes across the entire paper. This represents where land and sky meet.

2- Vanishing Point: This is where many of our lines will be directed in order to create that 3-D affect.

Where do I Begin?

Step 1- Now that we have our horizon line and vanishing point, we can start by drawing a road. Use a ruler to draw two lines that lead to the vanishing point, this should resemble a triangle.

Step 2-From here we can start to draw a building by creating two straight lines that are perpendicular to our horizon line.

Math Behind Perspective Drawing

Step 3-Then line up the outermost corner of the building with the vanishing point using a ruler, and draw a line. Do this with each corner of our rectangle for a 3-D effect.

Math Behind Perspective Drawing

Step 3- continued….

Math Behind Perspective Drawing

Step 4- For the remaining lines, use parallel and perpendicular lines to finish off our building.

Math Behind Perspective Drawing

Step 5- Get creative! Add more buildings, windows, antennas, and anything else you might see in a city -scape. Use your imagination! ๐Ÿ™‚

This method of perspective drawing is called one-point perspective because there is one vanishing point. But there are also 2-point and 3-point perspectives we can draw!

Want to learn how to do 2-Point Perspective drawing with 2 vanishing points!? Check out the video above to see how geometry and angles are related to this technique of perspective drawing!

Still got questions or want to learn more about perspective drawing? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! ๐Ÿ™‚

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For more math + art, check out this post on fractals found in nature here.

How to Construct a 45 Degree Angle with a Compass

Greeting math friends and welcome to another wonderful week of MathSux! In today’s post we are going to break down how to construct a 45 degree angle with a compass. We will take this step by step starting with a simple straight edge, then we will create a 90 degree angle, and finally we will bisect that 90 degree angle to get two 45 degree angles. If you have any questions please don’t hesitate to check out the video and step-by-step GIF below. Thanks so much for stopping by and happy calculating! ๐Ÿ™‚

How to Construct a 45 degree Angle with a Compass

How to Construct a 45 Degree Angle with a Compass:

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 straight edge, connect this intersection to point A. Notice this forms a 90ยบ angle.

Step 8: Now to bisect our newly made 90ยบ 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 straight edge, draw a line from point A to the new intersection of arcs we just made.

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

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! ๐Ÿ™‚

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Looking for more constructions? Check out how to construct a square inscribed in a circle and an equilateral triangle by clicking on their respective links!

How to find the Area of a Parallelogram: Geometry

Hey math peeps! In today’s post, we are going to go over how to find the area of a parallelogram. There is an easy formula to remember, A=bh, but we are going to look at why this formula works in the first place and then solve a few examples. Just a quick warning: The following examples do use special triangles and if you are need of a review, check out the posts here for 45 45 90 and 30 60 90 special triangles. Also, don’t forget to watch the video and try the practice problems below. Thanks so much for stopping by and happy calculating!

Area of a Parallelogram Formula:

How to find the Area of a Parallelogram

Why does the Formula for Area of a Parallelogram work?

Did you notice that the formula for area of a parallelogram above, base times height, is the same as the area formula for a rectangle?  Why?

If we cut off the triangle that naturally forms along the dotted line of our parallelogram, rotated it, and placed it on the other side of our parallelogram, it would naturally fit like a puzzle piece and create a rectangle! Check it out below:

How to find the Area of a Parallelogram

Now that we know where this formula comes from, let’s see it in action in the examples below:

Example #1:

How to find the Area of a Parallelogram

Step 1: Write out the formula:

Step 2:  Fill in the formula with values found on our parallelogram, b=12 inches h=4 inches, and multiply them together to get 48 inches squared.

How to find the Area of a Parallelogram

That was a simple example, but lets try a harder one that involves special triangles.

Example #2:

How to find the Area of a Parallelogram

Step 1: Write out the formula:

Step 2:  Label the values found on our parallelogram, b=10 ft and notice that we are going to need to find the value of the height.

Step 3: In order to find the value of the height, we need to remember our special triangles! We are not given the value of the height, but we are given some value of the triangle that is formed by the dotted line.  Let us take a closer look and expand this triangle:

Step 4: We can add in the missing 45ยบ degree value so that our triangle now sums to 180ยบ.

Step 5: Remember 45 45 90 special triangles(If you need a review click the link). Because that is exactly what we are going to need to find the value of the height! Below is our triangle on the left, and on the right is the 45 45 90 triangle ratios we need to know to find the value of the height.

Based on the above ratios, we can figure out that the height value is the same value as the base of the triangle, 2.

Step 6: If we place our triangle back into the original parallelogram, we can plug in our value for the height, h=2, into our formula to find the area:

How to find the Area of a Parallelogram

When you’re ready, check out the practice questions below!

Practice Questions:

Find the area of each parallelogram:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! ๐Ÿ™‚

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Constructing a Perpendicular Bisector

Hi everyone and welcome to MathSux! In this post we are going to be constructing a perpendicular bisector, a line that cuts a line segment in half and creates four 90ยบย angles. It’s a super fast and super simple construction! If you’re looking for more constructions, don’t forget to check more out here. Thanks so much for stopping by and happy calculating! ๐Ÿ™‚

What is the Perpendicular Bisector of a line ?

  • Cutโ€™s our line AB in half at its midpoint, creating two equal halves.
  • This will also create four 90ยบ angles about the line.
Constructing a Perpendicular Bisector

What is happening in this GIF?

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 straight edge.

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 90ยบ angles.

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! ๐Ÿ™‚

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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!

Inscribed Angles & Intercepted Arcs: Geometry

Ahoy math friends and welcome to MathSux! In this post, we are going to go over inscribed angles and intercepted arcs. We’ll break down the main basic rule for inscribed angles and the three theorems associated with this rule. If you are looking for more circle theorems, check out these posts on the Intersecting Secants Theorem and Central Angles Theorem. Also, don’t forget to check out the video and practice questions to truly master the topic below. Happy calculating! ๐Ÿ™‚

Inscribed Angles:

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.

Inscribed Angles & Intercepted Arcs

Inscribed Angle Theorems:

There are three inscribed angle theorems to know based on the rule stated above, check them out below!

Theorem #1: (Intercepted Arcs) In a circle when inscribed angles intercept the same arc, the angles are congruent.

Inscribed Angles & Intercepted Arcs

Theorem #2: In a circle when an angle is inscribed by a semicircle, it forms a  90ยบ angle.

Theorem #3: When a quadrilateral is inscribed in a circle, opposite angles are supplementary (add to 180ยบ). (The proof below shows angles A and C as supplementary, but this proof would also work for opposite angles B and D).

Inscribed Angles & Intercepted Arcs

Letโ€™s look at how to apply these rules with an Example:

a) Step 1: To find the value of angle CDB we need to look at our given information. We know that angle CAB=85ยบ, notice that this follows theorem number 3, โ€œWhen a quadrilateral is inscribed in a circle, opposite angles are supplementary.โ€ Therefore, we must subtract 110ยบ from 180ยบ to find the value of angle CDB.

b) Step 2: For finding angle ABD, weโ€™re going to use the same theorem we used in part a, opposite supplementary angles of an inscribed quadrilateral are supplementary.

c) Step 3: Next, to find the value of arc ABD, we need to use the basic inscribed angle theorem that tells us an inscribed angle is equal to half the value of its arc. Then use some basic algebra to solve for arc ABD.

d) Step 4: To find arc ACD, we need to use the basic inscribed angle theorem that tells us an inscribed angle is equal to  the value of its arc, then use algebra to solve similar to part c.

If this looks confusing, check out the video above! And when you are ready master this topic with the practice questions below!

Practice Questions:

Solutions:

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! ๐Ÿ™‚

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