Simultaneous Equations: Algebra

Happy new year and welcome to Math Sux! In this post we are going to dive right into simultaneous equations and how to solve them three different ways! We will go over how to solve simultaneous equations using the (1) Substitution Method (2) Elimination Method and (3) Graphing Method. Each and every method leading us to the same exact answer! At the end of this post don’t forget to try the practice questions choosing the method that best works for you! Happy calculating! 🙂

What are Simultaneous Equations?

Simultaneous Equations are when two equations are graphed on a coordinate plane and they intersect at, at least one point.  The coordinate point of intersection for both equation is the answer we are trying to find when solving for simultaneous equations. There are three different methods for finding this answer:

We’re going to go over each method for solving simultaneous equations step by step with the example below:

Method #1: Substitution

The idea behind Substitution, is to solve for 1 variable first algebraically, and the plug this value back into the other equation solving for one variable.  Then solving for the remaining variable.  If this sounds confusing, don’t worry! We’re going to do this step by step:

Step 1: Let’s choose the first equation and move our terms around to solve for y.

Simultaneous Equations

Step 3: The equation is set up and ready to solve for x!

Simultaneous Equations

Step 4: All we need to do now, is plug x=3 into one of our original equations to solve for y.

Simultaneous Equations

Step 5: Now that we have solved for both x and y, we have officially found where these two simultaneous equations meet!

Simultaneous Equations

Method #2: Elimination

The main idea of Elimination is to add our two equations together to cancel out one of the variables, allowing us to solve for the remaining variable.  We do this by lining up both equations one on top of the other and adding them together.  If variables at first do not easily cancel out, we then multiply one of the equations by a number so it can. Check out how it’s done step by step below!

Step 1: First, let’s stack both equations one on top of the other to see if we can cancel anything out:

Simultaneous Equations

Step 2: Our goal is to get a 2 in front of y in the first equation, so we are going to multiply the entire first equation by 2.

Simultaneous Equations

Step 3: Now that we multiplied the entire first equation by 2, we can line up our two equations again, adding them together, this time canceling out the variable y to solve for x.

Simultaneous Equations

Step 4: Now, that we’ve found the value of variable x=3, we can plug this into one of our equations and solve for missing unknown variable y.

Simultaneous Equations

Step 5: Now that we have solved for both x and y, we have officially found where these two simultaneous equations meet!

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Method #3: Graphing

The main idea of Graphing is to graph each a equation on a coordinate plane and then see at what point they intersect.  This is the best method to visualize and check our answer!

Step 1: Before we start graphing let’s convert each equation into y=mx+b (equation of a line) form.

Equation 1:

Equation 2:

Step 2: Now, let’s graph each line, y=3x-4 and y=-x+8, to see at what coordinate point they intersect.

Simultaneous Equations

Need to review how to draw an equation of a line? Check out this post here! Notice we got the same exact answer using all three methods (1) Substitution (2) Elimination and (3) Graphing.

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

Practice Questions:

Solve the following simultaneous equations for x and y.

Solutions:

  1. (1, 3)
  2. (4,5)
  3. (-1, -6)
  4. (3, -3)

Want more MathSux?  Don’t forget to check out our Youtube channel and more below! And if you have any questions, please don’t hesitate to comment below. Happy Calculating!

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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! 🙂

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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!

30 60 90 Special Triangles

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.

30 60 90 Special Triangles
30 60 90 Special Triangles

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

30 60 90 Special Triangles
30 60 90 Special Triangles

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!

30 60 90 triangle side lengths

How do I use this ratio?

30 60 90 triangle side lengths

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:

30 60 90 triangle side lengths

-> First let’s look at our ratio and compare it to our given triangle.

30 60 90 triangle side lengths

->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.

30 60 90 triangle side lengths
30 60 90 triangle side lengths

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.

30 60 90 triangle side lengths

->First let’s look at our ratio and compare it to our given triangle.

30 60 90 triangle side lengths

-> 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:

30 60 90 triangle side lengths
30 60 90 triangle side lengths

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

30 60 90 triangle side lengths

->First let’s look at our ratio and compare it to our given triangle.

30 60 90 triangle side lengths

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

30 60 90 triangle side lengths

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! 🙂

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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.

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:

NYE Ball Fun Facts

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.

NYE Ball Fun Facts

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.

NYE Ball Fun Facts

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:

NYE Ball Fun Facts

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!

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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?

Trig functions

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.

Trig functions

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

Trig functions

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. 

Trig functions

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).

Trig functions

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.

Trig functions

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! 🙂

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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! 

Graphing Trig Functions

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.

Graphing Trig Functions

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

Graphing Trig Functions

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! 

Graphing Trig Functions

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.

Graphing Trig Functions

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

Graphing Trig Functions

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! 

Graphing Trig Functions

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! 🙂

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Geometric Sequences: Algebra

Hi everyone and welcome to Mathsux! In this post, we’re going to go over geometric sequences. We’ll see what geometric sequences are, breakdown their formula, and solve two different types of examples. As always if you want more questions, check out the video below and the practice problems at the end of this post. Happy calculating! 🙂

What are Geometric Sequences?

Geometric sequences are a sequence of numbers that form a pattern when the same number is either multiplied or divided to each subsequent term.

Example:

geometric sequences

Notice we are multiplying 2 to each term in the sequence above. If the pattern were to continue, the next term of the sequence above would be 64. This is a geometric sequence!

In this sequence it’s easy to see what the next term is, but what if we wanted to know the 15th term?  That’s where the Geometric Sequence formula comes in!

Geometric Sequence Formula:

geometric sequences

Now that we broke down our geometric sequence formula, let’s try to answer our original question below:

->First, let’s write out the formula:

geometric sequences

-> Now let’s fill in our formula and solve with the given values.

geometric sequences

Let’s look at another example where, the common ratio is a bit different, and we are dividing the same number from each subsequent term:

-> First let’s identify the common ratio between each number in the sequence. Notice each term in the sequence is being divided by 2 (or multiplied by 1/2 ).

geometric sequences

-> Now let’s write out our formula:

geometric sequences

-> Next let’s fill in our formula and solve with the given values.

Practice Questions:

  1. Find the 12th term given the following sequence: 1250, 625, 312.5, 156.25, 78.125, ….
  2. Find the 17th term given the following sequence: 3, 9, 27, 81, 243,…..
  3. Find the 10th term given the following sequence: 5000, 1250, 312.5, 78.125 …..
  4. Shirley has $100 that she deposits in the bank. She continues to deposit twice the amount of money every month. How much money will she deposit in the twelfth month at the end of the year?

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 arithmetic sequences click this link here!

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Rotations about a Point: Geometry

rotations about a point

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! 🙂

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 90º,180º, 270º, -90º, -180º, or -270º.

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

rotations 90 degrees

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, it’s memorizing but if you need more options check out numbers 1 and 2 above!

Rotation Rules:

rotations 90 degrees

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 90º 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:

rotations

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. Let’s take a look at the Examples below:

Example #1:

rotations

Step 1: First, let’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, let’s draw an x and y axes starting at this point:

rotations

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

Step 2: Next, let’s take a look at our rule for rotating a coordinate -90º and apply it to our newly rotated triangles coordinates:

rotations

Step 3: Now let’s graph our newly found coordinate points for our new triangle .

rotations about a point

Step 4: Finally let’s connect all our new coordinates to form our solution:

rotations about a point

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

Example #2:

rotations about a point

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

rotations about a point

Step 2: Next we need to identify the direction of rotation.  Since we are rotating Point M 90º, 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.

rotations about a point

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

rotations about a point

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

Practice Questions:

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

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

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

rotations about a point

4.  Segment AB is rotated 270º 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! 🙂

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

The Unit Circle: Algebra 2/Trig.

Greetings math friends! In today’s post we’re going to go over some unit circle basics. We will find the value of trigonometric functions by using the unit circle and our knowledge of special triangles. For even more practice questions and detailed info., don’t forget to check out the video and examples at the end of this post. Keep learning and happy calculating! 🙂

What is the Unit Circle?

The Unit Circle is a circle where each point is 1 unit away from the origin (0,0).  We use it as a reference to help us find the value of trigonometric functions.

The Unit Circle

Notice the following things about the unit circle above:

  1. Degrees follow a counter-clockwise pattern from 0 to 360 degrees.
  2. Values of cosine are represented by x-coordinates.
  3. Values of sine are represented by y-coordinates.
  4. Using the unit circle we can find the degree and radian value of trigonometric functions (SOH CAH TOA). Check out the example below!

What’s the big deal with Quadrants?

Within a coordinate plane there are 4 quadrants numbered I, II, III, and IV used throughout all of mathematics. Within these quadrants there are different trigonometric functions that are positive to each unique quadrant.  This will be important when solving questions with reference angles later in this post. Check out which trig functions are positive in each quadrant below:

The Unit Circle
The Unit Circle

Now let’s look at some examples on how to find trigonometric functions using our circle!

The Unit Circle

Negative Degree Values:

The unit circle also allows us to find negative degree values which run clockwise, check it out below!

The Unit Circle

Knowing that negative degrees run clockwise, we can now find the value of trigonometric functions with negative degree values.

The Unit Circle

How to find trig ratios with 30º, 45º and 60º ?

Instead of memorizing much, much more of the unit circle, there’s a trick to memorizing two simple special triangles for answering these types of questions. The 45º 45º 90º  special triangle and the  30º 60º 90º special triangle. (Why does this work? These special triangles can also be derived and found on the unit circle).

special triangles

Using the above triangles and some basic trigonometry in conjugation with the unit circle, we can find so many more angles, take a look at the example below:

Since we need to find the value of tan(45º) , we will use the 45º, 45º, 90º  special triangle.

special triangles

For our last question, we are going to need to combine our knowledge of unit circles and special triangles:

-> In order to do this, we must first look at where our angle falls on the unit circle.  Notice that the angle 135º is encompassed by the pink lines and falls in quadrant 2.

The Unit Circle

-> Since our angle falls in the second quadrant where only the trig function sin is positive.  Since we are finding an angle with the function cosine, we know the solution will be negative.

-> Now we need to find something called a reference angle.  Which is what those θ, 180°-θ, θ-180°, 360°-θ and  symbols represent towards the center of the unit circle.  Using these symbols will help us find the value of cos(135º). 

Because the angle we are trying to find,135º , falls in the second quadrant, that means we are going to use the reference angle that falls in that quadrant 180º-θ theta, using the angle we are given as θ.

The Unit Circle

-> Now we can re-write and solve our trig equations using our newly found reference angle, 45º.

Now we are going to use our  45º 45º 90º special triangle and SOH CAH TOA to evaluate our trig function.  For a review on how to use SOH CAH TOA, check out this link here.

special triangles

When you’re ready, try the problems on your own below!

Practice Questions:

Solve the following trig functions using a unit circle and your knowledge of special triangles:

The Unit Circle

Solutions:

The Unit Circle

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above for even more examples. Happy calculating! 🙂

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If you’re looking to learn about how to draw trigonometric functions, check out his post here!

Arithmetic Sequences: Algebra

Hi everyone and welcome to Mathsux! In this post, we’re going to go over arithmetic sequences. We’ll see what arithmetic sequences are, breakdown their formula, and solve two different types of examples. As always if you want more questions, check out the video below and the practice problems at the end of this post. Happy calculating! 🙂

What are Arithmetic Sequences?

Arithmetic sequences are a sequence of numbers that form a pattern when the same number is either added or subtracted to each subsequent term.

Example:

arithmetic sequences

Notice we are adding 2 to each term in the sequence above. If the pattern were to continue, the next term of the sequence above would be 12. This is an arithmetic sequence!

In the above sequence it’s easy to see what the next term is, but what if we wanted to know the 123rd term?  That’s where the Arithmetic Sequence Formula comes in!

Arithmetic Sequence Formula:

arithmetic sequences

Now that we know the arithmetic sequence formula, let’s try to answer our original question below:

arithmetic sequence examples

-> First, let’s write the arithmetic sequence formula:

arithmetic sequences

-> Fill in our formula and solve with the given values.

Now let’s look at another example where we subtract the same number from each term in the sequence, making the common difference negative.

arithmetic sequence examples

-> First let’s identify the common difference between each number in the sequence. Notice each term in the sequence is being subtracted by 3.

arithmetic sequences

-> Now let’s write out our formula:

arithmetic sequences

-> Next let’s fill in our formula and solve with the given values.

Practice Questions:

  1. Find the 123rd term given the following sequence: 8, 12, 16, 20, 24, ….
  2. Find the 117th term given the following sequence: 2, 2.5, 3, 3.5, …..
  3. Find the 52nd term given the following sequence: 302, 300, 298, …..
  4. A software engineer charges $100 for the first hour of consulting and $50 for each additional hour.  How much would 500 hours of consultation cost?

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’re looking to learn more about sequences, check out these posts on Geometric Sequences and Recursive Formulas!

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Geometry: 45º 45º 90º Special Triangles

45 45 90 triangle

Greetings math folks! In this post we are going to go over 45º 45º 90º 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! 🙂

Why is it “special”?

 The 45º 45º 90º 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!

45 45 90 triangle

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 45º 45º 90º triangles!

45 45 90 triangle

How do I use this ratio?

45 45 90 triangle

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

Let’s try an example:

45 45 90 triangle
45 45 90 triangle sides
45 45 90 triangle sides
45 45 90 triangle sides
45 45 90 triangle sides
45 45 90 triangle sides

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.

45 45 90 triangle sides
45 45 90 triangle formula
45 45 90 triangle formula
45 45 90 triangle formula
45 45 90 triangle formula
45 45 90 triangle formula
45 45 90 triangle formula

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

Practice Questions: Find the value of the missing sides.

45 45 90 triangle formula
45 45 90 triangle formula

Solutions:

45 45 90 triangle formula

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

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