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# 7.6: Special Right Triangles

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Square Rock Garden

Ms. Kino’s class decided to do a community service project that everyone could enjoy. They decided to create a meditation garden that would be a rock garden.

Chas and Juanita took charge of the project. They drew a sketch of the rock garden and the rest of the class loved it so much that they instantly agreed to use the sketch that the pair had created. Here is their sketch.

“Let’s put a diagonal path in it,” Frankie suggested looking at the sketch.

“That’s a great idea, how long will the path be?” Chas asked.

The class wants to add a diagonal path. If they do that from one corner to another, how long will the path be?

This lesson will teach you all that you need to know to solve this problem. You will have a chance to solve it at the end of the lesson.

What You Will Learn

By the end of this lesson, you will have an understanding of the following skills.

• Recognize that a \begin{align*}45^\circ-45^\circ-90^\circ\end{align*} triangle is an isosceles triangle and that the length of the hypotenuse is the product of the length of either leg and \begin{align*}\sqrt{2}\end{align*}.
• Recognize that a \begin{align*}30^\circ-60^\circ-90^\circ\end{align*} triangle is half of an equilateral triangle, that the length of the hypotenuse is twice the length of the shorter leg, and that the longer leg is the product of the shorter leg and \begin{align*}\sqrt{3}\end{align*}.
• Find exact values of the indicated variable lengths in special right triangles, and check using the Pythagorean Theorem.
• Solve real – world problems involving applications of special right triangles.

Teaching Time

I. Understanding \begin{align*}\underline{45^\circ-45^\circ-90^\circ}\end{align*} Triangles

There are a few types of right triangles it is particularly important to study. Their sides are always in the same ratio, and it is crucial to study the \begin{align*}45^\circ-45^\circ-90^\circ\end{align*} and the \begin{align*}30^\circ-60^\circ-90^\circ\end{align*} triangles and understand the relationships between the sides. It will save you time and energy as you work through math problems both straight-forward and complicated. Let’s start by learning about the \begin{align*}45^\circ-45^\circ-90^\circ\end{align*}.

First, think about that \begin{align*}45^\circ-45^\circ-90^\circ\end{align*} refers to. Those values refer to the angle measures in the right triangle. We can see that there is one 90 degree angle and that the other two angles have the same measure. This particular triangle is also isosceles. An isosceles triangle has two side lengths that are the same. An isosceles right triangle will always have the same angle measurements: \begin{align*}45^\circ,45^\circ\end{align*}, and \begin{align*}90^\circ\end{align*} and will always have two side lengths that are the same. These characteristics make it a special right triangle.

Because these angles will always remain the same, the sides will always be in proportion. To find the relationship between the sides, use the Pythagorean Theorem.

Example

The isosceles right triangle below has legs measuring 1 centimeter. Use the Pythagorean Theorem to find the length of the hypotenuse.

As the problem states, you can use the Pythagorean Theorem to find the length of the hypotenuse. Since the legs are 1 centimeter each, set both \begin{align*}a\end{align*} and \begin{align*}b\end{align*} equal to 1 and solve for \begin{align*}c\end{align*}.

\begin{align*}a^2+b^2 &=c^2\\ (1)^2+(1)^2 &= c^2\\ 1+1 &= c^2\\ 2 &= c^2\\ \sqrt{2} &= \sqrt{c^2}\\ \sqrt{2} &= c\end{align*}

We can look at this and understand that there is also a 1 in front of the square root of two. This shows that the relationship between one side length and the length of the hypotenuse will always be the same. The hypotenuse of an isosceles right triangle will always equal the product of one leg and \begin{align*}\sqrt{2}\end{align*}.

Write this down in your notebook under \begin{align*}45^\circ-45^\circ-90^\circ\end{align*} special right triangles.

Now let’s use this information to solve the problem in the next example.

Example

What is the length of the hypotenuse in the triangle below?

Since the length of the hypotenuse is the product of one leg and \begin{align*}\sqrt{2}\end{align*}, you can easily calculate this length. It is easy because we know that with any 45/45/90 degree triangle, that the hypotenuse is the product of one of the legs and the square root of 2.

One leg is 3 inches, so the hypotenuse will be \begin{align*}3 \sqrt{2}\end{align*}, or about 4.24 inches.

That is a good question. To get that answer, we took the square root of two on the calculator, 1.414 and then multiplied it times 3.

\begin{align*}3 \times 1.414 = 4.242\end{align*}

We rounded to get the answer.

II. Understanding \begin{align*}\underline{30^\circ-60^\circ-90^\circ}\end{align*} Triangles

Another important type of right triangle has angles measuring \begin{align*}30^\circ,60^\circ\end{align*}, and \begin{align*}90^\circ\end{align*}. These triangles are exactly one half of equilateral triangles. Do you remember what an equilateral triangle is? An equilateral triangle is a triangle with equal angle measures. The equal angle measures of an equilateral triangle are \begin{align*}60^\circ-60^\circ-60^\circ\end{align*}. If we divide an equilateral triangle in half, then we have a \begin{align*}30^\circ-60^\circ-90^\circ\end{align*} triangle.

You can tell in the diagram that since the original triangle was equilateral, the short leg will be one-half the length of the hypotenuse.

Example

Find the length of the missing leg in the triangle below. Use the Pythagorean theorem to find your answer.

Just like you did for \begin{align*}45^\circ-45^\circ-90^\circ\end{align*} triangles, use the Pythagorean Theorem to find the missing side. In this diagram, you are given two measurements. The hypotenuse \begin{align*}(c)\end{align*} is 2 feet and the shorter leg \begin{align*}(a)\end{align*} is 1 foot. Find the length of the missing leg \begin{align*}(b)\end{align*}.

\begin{align*}a^2+b^2 &= c^2\\ (1)^2+b^2 &= (2)^2\\ 1+b^2 &=4\\ 1+b^2-1 &= 4-1\\ b^2 &= 3\\ \sqrt{b^2} &= \sqrt{3}\\ b &= \sqrt{3}\end{align*}

You can leave the answer as the radical as shown, or use your calculator to find the approximate value of 1.732 feet.

So, just as there is a constant proportion relating the sides of the \begin{align*}45^\circ-45^\circ-90^\circ\end{align*} triangle, there is also one relating the sides of the \begin{align*}30^\circ-60^\circ-90^\circ\end{align*} triangle. The hypotenuse will always be twice the length of the shorter leg, and the longer leg is always the product of the shorter leg and \begin{align*}\sqrt{3}\end{align*}.

Write this information down under 30/60/90 degree right triangles.

Example

What is the length of the missing leg in the a 30/60/90 degree right triangle with a short leg of 5 and a hypotenuse of 10?

Since the length of the longer leg is the product of the shorter leg and \begin{align*}\sqrt{3}\end{align*}, you can easily calculate this length. The short leg is 5 inches, so the hypotenuse will be \begin{align*}5 \sqrt{3}\end{align*}, or about 8.66 inches.

III. Find Missing Values in Special Right Triangles

In this section you can find exact values of the indicated variable lengths in special right triangles, and check using the Pythagorean Theorem. You now know the special proportions of the side lengths of certain right triangles, now you can use this information to solve problems involving these special right triangles. If you use these relationships correctly, you can solve a lot of problems about right triangles without too much time on calculation.

Example

What is the length of one leg in the triangle below?

The first step in this problem is to identify the type of right triangle depicted. Since the length of both sides is \begin{align*}k\end{align*}, this is an isosceles right triangle. This makes it a 45/45/90 degree triangle.

So, the hypotenuse is the product of one leg and \begin{align*}\sqrt{2}\end{align*}. Set up an equation to find the length of \begin{align*}k\end{align*}.

\begin{align*}k \times \sqrt{2} &= 9 \sqrt{2}\\ k \times \sqrt{2} \div \sqrt{2} &= 9 \sqrt{2} \div \sqrt{2}\\ k &= 9\end{align*}

The length of one leg in this triangle is 9 yards.

Example

What is the length of one leg in the triangle below?

The first step in this problem is to identify the type of right triangle depicted. The angles show that this is a \begin{align*}30^\circ-60^\circ-90^\circ\end{align*} triangle. So, the longer leg is the product of one leg and \begin{align*}\sqrt{3}\end{align*}. The hypotenuse is twice the length of the shorter leg. The shorter leg is 4 meters.

The hypotenuse will be \begin{align*}4 \times 2\end{align*}, or 8 meters long.

Now we can look at how special right triangles appear in real – world situations.

IV. Solve Real – World Problems Involving Applications of Special Right Triangles

Just like any other mathematical concept, you may be tested on this material in a real-world context. Just translate the information into mathematical data and proceed as usual. No new tools or skills are required to solve these problems.

Example

The diagram below shows the shadow a flagpole casts at a certain time of day. If the height of the flagpole is \begin{align*}7 \sqrt{3} \ feet\end{align*}, what is the length of the hypotenuse of the triangle formed by the flagpole and its shadow?

The wording in this problem is complicated, but you only need to notice a few things. You can tell in the picture that this triangle has angles of \begin{align*}30^\circ,60^\circ\end{align*}, and \begin{align*}90^\circ\end{align*}. The actual flagpole is the longer leg in the triangle, so use the ratios in the diagrams above to find the length of the hypotenuse.

The longer leg is the product of the shorter leg and \begin{align*}\sqrt{3}\end{align*}. So, the length of the shorter leg will be 7 feet.

The hypotenuse in a \begin{align*}30^\circ-60^\circ-90^\circ\end{align*} triangle will always be twice the length of the shorter leg, so it will equal \begin{align*}7 \times 2\end{align*}, or 14 feet.

Now let’s apply what we have learned to the problem from the introduction.

## Real-Life Example Completed

The Square Rock Garden

Here is the original problem once again. Reread it and then answer the question about the length of the diagonal path.

Ms. Kino’s class decided to do a community service project that everyone could enjoy. They decided to create a meditation garden that would be a rock garden.

Chas and Juanita took charge of the project. They drew a sketch of the rock garden and the rest of the class loved it so much that they instantly agreed to use the sketch that the pair had created. Here is their sketch.

“Let’s put a diagonal path in it,” Frankie suggested looking at the sketch.

“That’s a great idea, how long will the path be?” Chas asked.

The class wants to add a diagonal path. If they do that from one corner to another, how long will the path be?

Now use what you have learned to figure out the length of the path.

Solution to Real – Life Example

The first step in a word problem of this nature is to add important information to the drawing. Because the problem asks you to find the length of a path from one corner to another, you should draw that path in.

Once you draw the diagonal path, you can tell that this is a triangle question. Because both legs of the triangle have the same measurement (10 feet), this is an isosceles right triangle. The angles in an isosceles right triangle are \begin{align*}45^\circ,45^\circ\end{align*}, and \begin{align*}90^\circ\end{align*}.

In an isosceles right triangle, the hypotenuse is always equal to the product of the length of one leg and \begin{align*}\sqrt{2}\end{align*}. So, the length of the path will be the product of 10 and \begin{align*}\sqrt{2}\end{align*}, or \begin{align*}10 \sqrt{2} \ feet\end{align*}. This value is approximately equal to 14.14 feet.

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Isosceles Triangle
a triangle with two sides the same length.
45/45/90 Triangle
a special right isosceles triangle.
Equilateral Triangle
a triangle with all three angles \begin{align*}60^\circ\end{align*}.
30/60/90 Triangle
a special right triangle that is created when an equilateral triangle is divided in half.

## Time to Practice

Directions: Find the missing hypotenuse in each \begin{align*}45^\circ-45^\circ-90^\circ\end{align*} triangle.

1. Length of each leg = 5
2. Length of each leg = 4
3. Length of each leg = 6
4. Length of each leg = 3
5. Length of each leg = 7

Directions: Now use a calculator to figure out the approximate value of each hypotenuse. You may round to the nearest hundredth.

1. \begin{align*}5 \sqrt{2}\end{align*}
2. \begin{align*}4 \sqrt{2}\end{align*}
3. \begin{align*}6 \sqrt{2}\end{align*}
4. \begin{align*}3 \sqrt{2}\end{align*}
5. \begin{align*}7 \sqrt{2}\end{align*}

Directions: Find the missing length of the longer leg in each \begin{align*}30^\circ-60^\circ-90^\circ\end{align*} triangle.

1. short leg = 3
2. short leg = 4
3. short leg = 2
4. short leg = 8
5. short leg = 10

Directions: Now use a calculator to figure out the approximate value of each longer leg. You may round your answer when necessary.

1. \begin{align*}3 \sqrt{3}\end{align*}
2. \begin{align*}4 \sqrt{3}\end{align*}
3. \begin{align*}2 \sqrt{3}\end{align*}
4. \begin{align*}8 \sqrt{3}\end{align*}
5. \begin{align*}10 \sqrt{3}\end{align*}

Directions: Use what you have learned to solve each problem.

1. Janie had construction paper cut into and equilateral triangle. She wants to cut it into two smaller congruent triangles. What will be the angle measurement of the triangles that result?
2. Madeleine has poster board in the shape of a square. She wants to cut two congruent triangles out of the poster board without leaving any leftovers. What will be the angle measurements of the triangles that result?
3. A square window has a diagonal of \begin{align*}2 \sqrt{2} \ feet\end{align*}. What is the length of one of its sides?
4. A square block of cheese is cut into two congruent wedges. If a side of the original block was 9 inches, how long is the diagonal cut?
5. Jerry wants to find the area of an equilateral triangle but only knows that the length of one side is 4 centimeters. What is the height of Jerry’s triangle?

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