## 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! 🙂

## What is an Angle Bisector?

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

## Angle Bisector 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.

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

*Please note that the above example bisects a 70º angle, but this construction method will work for an angle of any size!🙂

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! 🙂

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

## 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.

## What are the Basics of 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.

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.

Step 3- continued….

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

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

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! 🙂

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

## 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! 🙂

## 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 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.

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

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

Let’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=85º, notice that this follows theorem number 3, “When a quadrilateral is inscribed in a circle, opposite angles are supplementary.” Therefore, we must subtract 110º from 180º to find the value of angle CDB.

b) Step 2: For finding angle ABD, we’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!

## 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! 🙂

## How to Calculate Z-Score?: Statistics

Hi there and welcome to MathSux! In this post, we are going to explore how to calculate z-score and the normal distribution. We’ll do this by examining the normal curve and learning how to find probability finding z-score and using the mean, standard deviation, and specific data points. Fore more info and more MathSix don’t forget to check out the video and practice questions below. Happy calculating!

## What is a Normal Curve?

A normal curve is a bell shaped curve that shows the distribution of data evenly spread with respect to the mean. If you look at the normal curve below, the area under the curve shows all the possible probabilities of a certain data point occurring, notice the curve is higher towards the center mean, μ, and gets smaller as the distance from μ grows. The distance from μ is measured by the standard deviation, a unique unit of measurement that is specific to each group of data.

Mean: The mean always falls directly in the center of our normal curve. It is the average of our data, and always falls right in the middle.

Standard Deviation:  This value is used as a standard unit of measurement for the data, measuring the distance between each data point in relation to the mean throughout the entire data set. For a review on what standard deviation is and how to calculate it, check out this post here.

Now for our normal curve:

Notice half of the data is below the mean, μ, while the other half is above? The normal curve is symmetrical about the mean, μ!

## How to Calculate Z-Score?

Z Score can tell us at what percentile a certain point in the data set falls in relation to the rest of the mean by using the standard deviation as a unit of measurement.  If this sounds confusing, it’s ok! Take a look at the following formula:

We use the above formula in conjunction with  a z table which tells us the probability under the curve for a certain point.

## a) What percent of student scored below 500?

Step 1: First, let’s draw out our given information the mean=500, standard deviation=100, and the data point the question is asking for x=500 onto a normal curve. Notice that we want to find the value of the area under the curve shaded in pink.  This will tell us the percent of students that scored below 500.

Step 2: We need to find the z-score by, using the data point given to us x=500, the mean=500, and the standard deviation, sigma=100.

Step 3: Yes, we have a zero! Now we need to take our z table and line up our chart. Notice that the chart finds the probability for everything at the beginning of the normal curve and on.  This is perfect for answering our question!

Step 4: The table gives us our solution of .5000.  If we multiply .5000 times 100 it gives us the percent of students who scored below 500 at 50%.

## b) What percent of student scored above 620?

Step 1: First, let’s draw out our given information the mean=500, standard deviation=100, and the data point the question is asking for x=620 onto a normal curve. Notice that we want to find the value of the area under the curve shaded in pink.  This will tell us the percent of students that scored above 620.

Step 2: We need to find the z-score by, using the data point given to us x=620, the mean=500, and the standard deviation, sigma=100.

Step 3: Yes, we got 1.2! Now we need to take our z table and line up our chart. Notice that the chart finds the probability for everything at the beginning of the normal curve and on.  This is means to find the percent we are looking for, we need to subtract our answer from one since we want the value of probability on the right side of the curve (the z-table only provides the left side).

Step 4: The table gives us our solution of .8849.  If we subtract this value from 1 then multiply that value times 100 it gives us the percent of students who scored above 620.

## C) What is the highest score a student could receive if the students was in the 16.11th percentile?

Step 1: In this question we have to work backwards by first identifying, where on the z-score table is the number .1611 and then filling in our z score formula to find x, the missing data point (in this case test score).

Search the table for .1611:

Notice that .1611 can be found on the z-table above with z-score -0.99.  This is what we’ll use to find the unknown data point!

Step 2: We need to find the unknown test score by, using the z score we just found z=-0.99, the mean=500, and the standard deviation, sigma=100.

Step 3: Solve for x.

## Practice Questions:

The grades on a final English exam are normally distributed with a mean of 75 and a standard deviation of 10.

a) What percent of students scored below a 60?

b) What percent of students scored above an 89?

c) What is the highest possible grade that included in the 4.46th percentile?

d) What percent of students got at least a 77?

## Solutions:

a) 6.68%

b) 8.08%

c) 58

d) 42.07%

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

## 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! 🙂

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

## What is a 30 60 90 Triangle and why is it “Special”?

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 let’s draw a mirror image of our triangle.  Next, we can label the length of the new side opposite 30º “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.

Let’s try an Example:

-> First let’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 let’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 let’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 60º and need to find the other two missing sides.

->First let’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, let’s fill it in our triangle below:

Finally, let’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! 🙂

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 today’s post we’re going do something a little different and take a look at the math behind the very famous and very shiny New Year’s Eve Ball that drops down every year at midnight.  We’ll break down the shape, the volume, and the number of those dazzling Waterford crystals (and no this post isn’t sponsored) and look at some NYE Ball Fun Facts.

## Shape: Geodesic Sphere

Yes, apparently the shape of the New Year’s ball is officially called a “Geodesic Sphere.”  It is 12 feet in diameter and weighs 11,875 pounds.

## Volume (Estimate):  288π ft3

If we wanted to estimate the volume of the New Year’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 (LED’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: Let’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, you’ll want to find the volume of the Hudson Yards Vessal in NYC here.  Happy calculating and Happy New Year from MathSux!

## Trig Functions (Amp, Freq, Phase Shifts): Algebra 2/Trig.

Hi everyone, and welcome to MathSux! In this post we are going to break down how to graph trig functions by identifying its amplitude, frequency, period, and horizontal and vertical phase shifts. Fear not! Because we will breakdown what each of these mean and how to find them, then apply each of these changes step by step on our graph. And if you’re ready for more, check out the video and the practice problems below, happy calculating! 🙂

*For a review on how to derive basic Trig functions (y=sinx, y=cosx, and y=tanx), click here.

## What are the Different Parts of a Trig Function?

Amplitude: The distance (or absolute value) between the x-axis and the highest point on the graph.

Frequency: This is the number of cycles that happen between 0 and 2π. (Α “cycle” in this case is the number of “s” cycles for the sine function).

Period: The x-value/length of one cycle. (Α “cycle” in this case is the number of “s” cycles for the sine function). This is found by looking at the graph and seeing where the first cycle ends, or, by using the formula:

Horizontal Shift: When a trigonometric function is moved either left or right along the x-axis.

Vertical Shift: When a trigonometric function is moved either up or down along the y-axis.

Let’s try an Example, graphing a Trig Function step by step.

Step 1: First let’s label and identify all the different parts of our trig function.

Step 2: Now let’s transform our graph one step at a time.  First let’s start graphing y=cos(x) without any transformations.

Step 3: Let’s add our amplitude of 2, the distance to the x-axis.  To do this our highest and lowest points on the y-axis will now be moved to 2 and -2 respectively.

Step 4: Next, we do a horizontal phase shift to the left by (π/2).  To do this, we look at where negative (π/2) is on our graph at (-π/2) and move our entire graph over to start at this new point, “shifting” it to (π/2).

Step 5: For our last transformation, we have a vertical phase shift up 1 unit.  All this means is that we are going to shift our entire graph up by 1 unit along the y-axis.

## Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above for an in depth explanation. Happy calculating! 🙂

## Graphing Linear Inequalities: Algebra

graphing linear inequalities

Hi and welcome to MathSux! In this post, we are going to go over the rules for graphing linear inequalities on a coordinate plane when it comes to drawing lines, circles , and shading, then we are going to solve an example step by step. If you have any questions, check out the video below and try the practice questions at the end of this post! If you still have questions, don’t hesitate to comment below and happy calculating! 🙂

## Graphing Linear Inequalities:

When graphing linear inequalities, we always want to treat the inequality as an equation of a line in  form y=mx+b….with a few exceptions:

Now that we know the rules, of graphing inequalities, let’s take a look at an Example!

Step 1: First, let’s identify what type of inequality we have here.  Since we are working with a > sign, we will need to use a dotted line and open circles when creating our graph.

Step 2: Now we are going to start graphing our linear inequality as a normal equation of a line, by identifying the slope and the y-intercept only this time keeping open circles in mind.  (For a review on how to graph regular equation of a line in y=mx+b form, click here)

Step 3: Now let’s connect our dots, by using a dotted line to represent our greater than sign.

Step 4: Now it is time for us to shade our graph, since this is an inequality, we need to show all of our potential solutions with shading.  Since we have a greater than sign, , we will be shading above the y-axis.  Notice all the positive y-values above are included to the left of our line.  This is where we will shade.

Step 5: Check!  Now we need to check our work.  To do that, we can choose any point within our shaded region, if the coordinate point we chose hold true when plugged into our inequality then we are correct!

Let’s take the point (-3,2) plugging it into our inequality where x=-3 and y=2.

## Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above for an in depth explanation. Happy calculating! 🙂

## 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:

Let’s look at how to apply these rules with an Example:

Let’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! 🙂

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

## Graphing Trig Functions: Algebra 2/Trig.

Hi everyone and welcome to MathSux! This post is going to help you pass Algebra 2/Trig. In this post, we are going to apply our knowledge of the unit circle and trigonometry and apply it to graphing trig functions y=sin(x), y=cos(x), and y=tan(x). If you have any questions, don’t hesitate to comment or check out the video below. Thanks for stopping by and happy calculating! 🙂

## How do we get coordinate points for graphing Trig Functions?

For deriving our trigonometric function graphs [y=sin(x), y=cos(x), and y=tan(x)] we are going to write out our handy dandy Unit Circle. By looking at our unit circle and remembering that coordinate points are in (cos(x), sin(x)) form and that tanx=(sin(x))/(cos(x)) we will be able to derive each and every trig graph!

*Note below is the unit circle we are going to reference to find each value, for an in depth explanation of the unit circle, check out this link here.

## How to Graph y=sin(x)?

Step 1: We are going to derive each degree value for sin by looking at the unit circle. These will be our coordinates for graphing y=sin(x). *For a review on how to get these values, check out the link here explaining the unit circle.

Step 2: Now we need to convert all the  from degrees to radians.  Fear not because this can be done easily with a simple formula!

To convert degrees à radians, just use the formula below:

Step 3: Now that we have our coordinate points and converted degrees to radians, we can draw out our function y=sin(x) on the coordinate plane!

Now we will follow the same process for graphing y=cos(x) and y=tan(x).

## How to Graph y=cos(x)?

Step 1: We are going to derive each degree value for cos by looking at the unit circle. These will be our coordinates for graphing y=cos(x). *For a review on how to get these values, check out the link here explaining the unit circle.

Step 2: Now we need to convert all the  from degrees to radians.

Step 3: Now that we have our coordinate points and converted degrees to radians, we can draw out our function y=cos(x) on the coordinate plane!

## How to Graph y=tan(x)?

Step 1: We are going to derive each degree value for tan by looking at the unit circle. In order to derive values for tan(x), we need to remember that tan(x)=sin(x)/cos(x). Once found, these will be our coordinates for graphing y=tan(x). *For a review on how to get these values, check out the link here explaining the unit circle.

Step 2: Now we need to convert all the  from degrees to radians.

Step 3: Now that we have our coordinate points and converted degrees to radians, we can draw out our function y=tan(x) on the coordinate plane!

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above for an in depth explanation. Happy calculating! 🙂