## 30 60 90 Special Triangles: Geometry

Hi everyone and welcome to MathSux! In this post we are going to break down 30 60 90 degree special triangles. What is it? Where did it come from? What are the ratios of it’s side lengths and how to do we use them? You will find all of the answers to these questions below. Also, don’t forget to check out the video below and practice questions at the end of this post. Happy calculating! đź™‚

Want to make math suck just a little bit less? Subscribe to my Youtube channel for free math videos every week! đź™‚

## What is a 30 60 90 Triangle and why is it â€śSpecialâ€ť?

The 30 60 90 triangle is special because it forms an equilateral triangle when a mirror image of itself is drawn, meaning all sides are equal!  This allows us to find the ratio between each side of the triangle by using the Pythagorean theorem. Check it out below!

Now letâ€™s draw a mirror image of our triangle.  Next, we can label the length of the new side opposite 30Âş “a,” and add this new mirror image length with the original we had to get, a+a=2a.

If we look at our original 30 60 90 triangle, we now have the following values for each side based on our equilateral triangle:

Now we can re-label our triangle, knowing the length of the hypotenuse in relation to the two legs. This creates a ratio that applies to all 30 60 90 triangles!

## How do I use this ratio?

Knowing the above ratio, allows us to find any length of any and every 30 60 90 triangle, when given the value of one of its sides.

Letâ€™s try an Example:

-> First letâ€™s look at our ratio and compare it to our given triangle.

->Notice we are given the value of a, which equals 4, knowing this we can now fill in each length of our triangle based on the ratio of a 30 60 90 triangle.

Now letâ€™s look at an Example where we are given the length of the hypotenuse and need to find the values of the other two missing sides.

->First letâ€™s look at our ratio and compare it to our given triangle.

-> Notice we are given the value of the hypotenuse, 2a=20. Knowing this we can find the value of a by dividing 20 by 2 to get a=10. Once we have the value of a=10, we can easily find the length of the last leg based on the 30 60 90 ratio:

Now for our last Example, when we are given the side length across from 60Âş and need to find the other two missing sides.

->First letâ€™s look at our ratio and compare it to our given triangle.

-> In this case, we need to use little algebra to find the value of a, using the ratio for 30 60 90 triangles.

Now that we have one piece of the puzzle, the value of a, letâ€™s fill it in our triangle below:

Finally, letâ€™s find the value of the length of the hypotenuse, which is equal to 2a.

## Practice Questions:

Find the value of the missing sides of each 30 60 90 degree triangle.

## Solutions:

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

Looking to review 45 45 90 degree special triangles? Check out this post here!

## NYE Ball Fun Facts: Volume & Combinations

Greetings math friends, and Happy New Year! In todayâ€™s post weâ€™re going do something a little different and take a look at the math behind the very famous and very shiny New Yearâ€™s Eve Ball that drops down every year at midnight.  Weâ€™ll break down the shape, the volume, and the number of those dazzling Waterford crystals (and no this post isnâ€™t sponsored) and look at some NYE Ball Fun Facts.

## Shape: Geodesic Sphere

Yes, apparently the shape of the New Yearâ€™s ball is officially called a â€śGeodesic Sphere.â€ť  It is 12 feet in diameter and weighs 11,875 pounds.

## Volume (Estimate): 288Ď€ ft3

If we wanted to estimate the volume of the New Yearâ€™s Ball we would could use the formula for volume of a sphere:

## Number of Waterford Crystals: 2,688

Talk about the ultimate shiny bauble! The NYE ball lights up the night with all 2,688 crystals in the shape of different sized triangles, each with heights of 5.75 inches or 4.75 inches.

## Number of Lights: 48 light emitting diodes (LEDâ€™s)

On each triangle, there are 48 LEDs: 12 red, 12 blue, 12 green, and 12 white, for a total of 32,256 LEDs on the entire NYE ball itself.

## Permutations and Combinations:

Permutations: With this many lights and colors, there are over a billion potential permutations of colors on the entire NYE ball.

Combinations: Letâ€™s break down one triangle with 48 LED lights each with 12 red, 12 blue, 12 green, and 12 white LEDs. How many possible combinations of lights are possible if we were to choose 7 blue, 5 red, 10 green, and 1 white turned on all at the exact same time?

We end up with the combination formula below:

That means that there are 496,793,088 possible ways that 7 blue lights, 5 red lights, 10 green lights, and 1 white light can be lit up on a triangle that is part of the entire NYE ball!

Interested in more NYE fun facts?  Check out the sources of this article here.

NYE Fact Sheet from: timessquarenyc.org

NYE Ball picture: Timesquareball.net

If you like finding the volume of the NYE ball maybe, youâ€™ll want to find the volume of the Hudson Yards Vessal in NYC here.  Happy calculating and Happy New Year from MathSux!

## Central Angles Theorems: Geometry

Hi everyone, and welcome to MathSux! In this post, we are going to go over the Central Angles Theorems of circles. We’ll go over the theorems associated with central angles and then solve a quick example. Make sure to test your understanding of central angles and arcs with the practice questions at the end of this post. And, if you want more, don’t forget to check out the video below, happy calculating!

## Central Angles and Arcs:

Central angles and arcs form when two radii are drawn from the center point of a circle.  When these two radii come together they form a central angle. A central angle is equal to the length of the arc. When it comes to measuring the central angle, the central angle is always equal to arc length and vice versa:

Central Angles = Arc Length

## Central Angle Theorems:

There are a two central angle theorems to know, check them out below!

Central Angle Theorem #1:

Central Angle Theorem #2:

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

Letâ€™s do this one step at a time.

## Solutions:

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

*Also, if you want to check out Intersecting Secants click this link here!

## TikTok Math Video Compilations

Happy December everyone! With crazy 2020 coming to an end, I thought I would share some TikTok math video compilations of Algebra, Geometry, Algebra 2/Trig, and Statistics for a quick review of all our videos posted throughout the year. Enjoy these TikTok math video compilations and happy calculating! đź™‚

Want to make math suck just a little bit less? Subscribe and follow us for FREE fun colorful math videos and lessons every week! đź™‚

## Algebra:

Within algebra, you will find arithmetic sequences, combining like terms, box and whisker plots, geometric sequences, solving radical equations, completing the square, 4 ways to factor quadratic equations, piecewise functions and more!

## Geometry:

Within Geometry, you will find, how to construct an equilateral triangle, a median of a trapezoid, area of a sector, how to find perpendicular and parallel lines through a given point, SOH CAH TOA right triangle trigonometry, reflections, and more!

## Algebra 2/Trig.

Within Algebra 2/Trig., you will find, how to expand a cubed binomial, how to divide polynomials, how to solve log equations, imaginary numbers, synthetic division, unit circle basics, how to graph y=sin(x), and more!

## Statistics:

Within statistics, you will find, box and whisker plots, how to find the variance, and, the probability of flipping a coin 2 times!

For full length video, don’t forget to check out our free math video index page! Thanks for stopping by! đź™‚

## Rotations about a Point: Geometry

Happy Wednesday math friends! In this post we’re going to dive into rotations about a point! In this post we will be rotating points, segments, and shapes, learn the difference between clockwise and counterclockwise rotations, derive rotation rules, and even use a protractor and ruler to find rotated points. The fun doesn’t end there though, check out the video and practice questions below for even more! And as always happy calculating! đź™‚

## What are 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 90Âş,180Âş, 270Âş, -90Âş, -180Âş, or -270Âş.

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

## 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, itâ€™s memorizing but if you need more options check out numbers 1 and 2 above!

## 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 90Âş 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:

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. Letâ€™s take a look at the Examples below:

## Example #1:

Step 1: First, letâ€™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, letâ€™s draw an x and y axes starting at this point:

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

Step 2: Next, letâ€™s take a look at our rule for rotating a coordinate -90Âş and apply it to our newly rotated triangles coordinates:

Step 3: Now letâ€™s graph our newly found coordinate points for our new triangle .

Step 4: Finally letâ€™s connect all our new coordinates to form our solution:

Another type of question with rotations, may not involve the coordinate plane at all! Letâ€™s look at the next example:

## Example #2:

Step 1: First, letâ€™s identify the point we are rotating (Point M) and the point we are rotating about (Point K).

Step 2: Next we need to identify the direction of rotation.  Since we are rotating Point M 90Âş, we know we are going to be rotating this point to the left in the clockwise direction.

Step 3: Now we can draw a line from the point of origin, Point K, to Point M.

Step 4: Now, using a protractor and ruler, measure out 90Âş, draw a line, and notice that point L lands on our 90Âş line. This is our solution! (Note: For help on how to use a protractor, check out the video above).

Ready for more? Check out the practice questions below to master your rotation skills!

## Practice Questions:

1. Point B is rotated -90Âş about the origin. Which point represents newly rotated point B?

2. Triangle ABC is rotated -270Âş about point M.  Show newly rotated triangle ABC as A prime B prime C prime.

3. Point G is rotated about point B by 180Âş. Which point represents newly rotated point B?

4.  Segment AB is rotated 270Âş about point K.  Show newly rotated segment AB.

## Solutions:

Still got questions on how to rotations about a point? No problem! Don’t hesitate to comment with any questions or check out the video above for even more examples. Happy calculating! đź™‚

Looking to brush up on your rotations skills? Check out this post here!

## Geometry: 45Âş 45Âş 90Âş Special Triangles

45 45 90 triangle

Greetings math folks! In this post we are going to go over 45Âş 45Âş 90Âş special triangles and how to find the missing sides when given only one of its lengths. For even more examples, check out the video below and happy calculating! đź™‚

## Why is it â€śspecialâ€ť?

The 45Âş 45Âş 90Âş triangle is special because it is an isosceles triangle, meaning it has two equal sides (marked in blue below).  If we know that the triangle has two equal lengths, we can find the value of the hypotenuse by using the Pythagorean Theorem.  Check it out below!

Now we can re-label our triangle, knowing the length of the hypotenuse in relation to the two equal legs. This creates a ratio that applies to all 45Âş 45Âş 90Âş triangles!

## How do I use this ratio?

Knowing the above ratio, allows us to find any length of a 45Âş 45Âş 90Âş triangle, when given the value of one of its sides.

Letâ€™s try an example:

Now letâ€™s look at an example where we are given the length of the hypotenuse and need to find the values of the other two missing sides.

Now try mastering the art of the 45Âş 45Âş 90Âş special triangle on your own!

## Solutions:

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

## Transversals and Parallel Lines: Geometry

Happy Wednesday math friends! In this post we are going to look at transversals and parallel lines and find the oh so many congruent and supplementary angles they form when they come together! Congruent angles that form with these types of lines are more commonly known as Alternate Interior Angles, Alternate Exterior Angles, Corresponding angles, and Supplementary angles. Let’s look at this one step at a time:

## What are Transversals and Parallel Lines?

When two parallel lines are cut by a diagonal line ( called a transversal) it looks something like this:

Each angle above has at least one congruent counterpart. There are several different types of congruent relationships that happen when a transversal cuts two parallel lines and we are going to break each down:

## 1) Alternate Interior Angles:

When a transversal line cuts across two parallel lines, opposite interior angles are congruent.

## 2) Alternate Exterior Angles:

When a transversal line cuts across two parallel lines, opposite exterior angles are congruent.

## 3) Corresponding Angles:

When a transversal line cuts across two parallel lines, corresponding angles are congruent.

## 4) Supplementary Angles:

Supplementary angles are a pair of angles that add to 180 degrees. 180 degrees is the value of distance found within a straight line, which is why you’ll find so many supplementary angles below:

Knowing the different sets of congruent and supplementary angles, we can easily find any missing angle values when faced with the following question:

-> Using our knowledge of congruent and supplementary angles we should be able to figure this out! Right away we can find angle 2 by noticing angle 1 and angle 2 are supplementary angles (add to 180 degrees).

-> Knowing angle 2 is 50 degrees, we can now fill in the rest of our transversal angles based on our corresponding and supplementary rules.

Try the following transversal and parallel lines questions below! Some may a bit harder than the previous example, if you get stuck, check out the video that goes over a similar example above and happy calculating! đź™‚

Practice Questions:

1. Find the value of the missing angles given line r  is parallel to line  s and line t is a transversal.

2. Find the value of the missing angles given line r is parallel to line s and line t is a transversal.

Solutions:

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

Also, if you’re looking to learn about the difference between parallel and perpendicular lines, check out this post here!

## 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 a ruler. For anyone new to constructions, this is the perfect topic for art aficionados since there is more drawing than there is actual math.

## What is an Equilateral Triangle?

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:

Now, for our Example:

## Whatâ€™s Happening in this GIF?

1. Using a compass, measure out the distance of line segment  .

2. With the compass on point A, draw an arc that has the same distance as .

3. With the compass on point B, draw an arc that has the same distance as .

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!

Still got questions? No problem! Don’t hesitate to comment with any questions. Happy calculating! đź™‚

## Reflections: Geometry

Greetings and welcome to Mathsux! Today we are going to go over reflections, one of the many types of transformations that come up in geometry.  And thankfully, it is one of the easiest transformation types to master, especially if you’re more of a visual learner/artistic type person. So let’s get to it!

## What are 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.

## Example:

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

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, to reflect it we are going to count 1 unit to the left of the line x=-2 and label our new point, C|.

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|.

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|.

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

## Solutions:

Still got questions?  No problem! Check out the video above or comment below! Happy calculating! đź™‚

Looking to review rotations about a point? Check out this post here!

## Intersecting Secants Theorem: Geometry

Ahoy! Today weâ€™re going to cover the Intersecting Secants Theorem!  If you forgot what a secant is in the first place, donâ€™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, donâ€™t hesitate to comment below and check out my video for more of an explanation. Stay positive math peeps and happy calculating! đź™‚

Wait, what are Secants?

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:

Letâ€™s now see how we can apply the intersecting Secants Theorem to find missing length.

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

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

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!

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