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# 7.3: The Pythagorean Theorem

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Painting Project

While Mr. Kennedy’s class was working on the garden, the students in Ms. Richardson’s class decided to paint the equipment shed where all of the sports equipment was kept.

“That thing hasn’t been painted in decades,” Karen said in the first meeting.

“I agree, it does look awful,” Cameron added.

“Well, I don’t know about decades, but it does need to be painted, so that is what we are going to do. Now to work on this project, we will need to choose a ladder that will reach up high enough. How can we figure this out?” Ms. Richardson asked.

The class was silent.

“I have an idea,” Veria said smiling. “It has to do with triangles. We need to figure out the height of the ladder, compared to the height of the building.”

“Yes, but don’t forget that the ladder reaches out from the building, not up against it. So we have to consider that measurement too,” Aran chimed in.

“What can we use to figure that out?” Karen asked.

Once again, the group was silent.

A mathematical formula is needed to solve this problem. The students have their work cut out for them. So do you. Pay attention to this lesson so that you can explain the formula that they will need and why they will need it.

What You Will Learn

By the end of this lesson, you will be able to perform the following skills.

• Derive the Pythagorean Theorem.
• Use the Pythagorean Theorem to find missing dimensions of right triangles.
• Derive the converse of the Pythagorean Theorem.
• Use the converse of the Pythagorean Theorem to identify right triangles.

Teaching Time

I. Derive the Pythagorean Theorem

You have probably already studied many different types of triangles. Acute triangles have angles that are all less than 90\begin{align*}90^{\circ}\end{align*}. Obtuse triangles have one angle that is between 90\begin{align*}90^{\circ}\end{align*} and 180\begin{align*}180^{\circ}\end{align*}. Right triangles have one angle that measures exactly 90\begin{align*}90^{\circ}\end{align*}—in other words, it has one right angle.

This lesson focuses entirely on properties specific to right triangles. While all of the equations and strategies you are about to learn are helpful, they apply only to right triangles – they will not work with acute or obtuse triangles.

To begin, let’s look at the parts of a right triangle.

The legs are the two sides of the triangle that are labeled a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*}. The hypotenuse is the longest side of a right triangle and it is labeled c\begin{align*}c\end{align*}. There is a special relationship between the legs of a right triangle and the hypotenuse of a right triangle.

One of the special characteristics of right triangles is described by the Pythagorean Theorem, thought to have been developed around 500 B.C.E. It states that the squared value of the hypotenuse will equal the sum of the squares of the two legs. In the triangle above, the sum of the squares of the legs is a2+b2\begin{align*}a^2 + b^2\end{align*} and the square of the hypotenuse is c2\begin{align*}c^2\end{align*}. So, the Pythagorean theorem is commonly represented as a2+b2=c2\begin{align*}a^2 + b^2 = c^2\end{align*} where a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*} are the legs of the right triangle and c\begin{align*}c\end{align*} is the hypotenuse.

The Pythagorean Theorem is known as a2+b2=c2\begin{align*}a^2 + b^2 = c^2\end{align*}.

You may be asking yourself why that is the case. Well, we can think about the Pythagorean Theorem in terms of a square. We know that there is a relationship between a square and a right triangle. We can divide a square with a diagonal and because a square has four right angles, the diagonal will divide the square into two right triangles. Now because a right triangle comes from the square, the sides will also be related to the square. This is where the Pythagorean Theorem comes from.

Let’s look at an example.

Example

Use the measures of the triangle below to test the Pythagorean theorem.

The legs of the triangle above are 3 inches and 4 inches. The hypotenuse is 5 inches. So, a=3\begin{align*}a = 3\end{align*}, b=4\begin{align*}b = 4\end{align*}, and c=5\begin{align*}c = 5\end{align*}. We can test the formula to see if this is true.

a2+b232+42(3×3)+(4×4)9+1625=c2=52=(5×5)=25=25\begin{align*}a^2 + b^2 &= c^2\\ 3^2 + 4^2 &= 5^2\\ (3 \times 3) + (4 \times 4) &= (5 \times 5)\\ 9 + 16 &= 25\\ 25 &= 25\end{align*}

Since both sides of the equation equal 25, the equation is true. Therefore, the Pythagorean theorem worked on this right triangle.

This combination of numbers (3, 4, 5) is referred to as a Pythagorean triple. In other words, these three numbers work together to make the Pythagorean Theorem true. Throughout this chapter, you will learn about other Pythagorean triples as well.

II. Use the Pythagorean Theorem to Find Missing Dimensions of Right Triangles

Now that you have learned how to derive and execute the Pythagorean Theorem, there are many different ways to apply it. Any time you have two out of three sides in a right triangle, you can find the third using the equation a2+b2=c2\begin{align*}a^2 + b^2 = c^2\end{align*}, where a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*} are the lengths of the legs of the triangle and c\begin{align*}c\end{align*} is the length of the hypotenuse.

When applying the Pythagorean Theorem, be sure to use exponents and square roots accurately.

Example

What is the length of b\begin{align*}b\end{align*} in the triangle below?

Use the Pythagorean Theorem to identify the length of the missing leg, b\begin{align*}b\end{align*}. Be sure to simplify the exponents and roots carefully. Also remember to use inverse operations to solve the equation properly.

a2+b2=c2,\begin{align*}a^2 + b^2 = c^2,\end{align*} where a=6\begin{align*}a = 6\end{align*} and c=10\begin{align*}c = 10\end{align*}

62+b236+b236+b236b2b2b=102=100=10036=64=64=8\begin{align*}6^2 + b^2 &= 10^2\\ 36 + b^2 &= 100\\ 36 + b^2 - 36 &= 100-36\\ b^2 &= 64\\ \sqrt{b^2} &= \sqrt{64}\\ b &= 8\end{align*}

The length of the missing side is 8 inches.

You already know about the Pythagorean triple 3:4:5. Notice that this triangle is proportional to that ratio. If you divide the lengths of the triangle in the example by two, you find the same proportion—3:4:5. Whenever you find a Pythagorean triple, you can apply those ratios with greater factors as well. So, 6, 8, 10 is another Pythagorean triple.

Example

Find the length of the missing side in the triangle below.

Use the Pythagorean Theorem to identify the length of the missing hypotenuse. Be sure to simplify the exponents and roots carefully. Also remember to use inverse operations to solve the equation properly.

a2+b2=c2,\begin{align*}a^2 + b^2 = c^2,\end{align*} where a=5\begin{align*}a = 5\end{align*} and b=12\begin{align*}b = 12\end{align*}

52+12225+14416916913=c2=c2=c2=c2=c\begin{align*}5^2 + 12^2 &= c^2\\ 25 + 144 &= c^2\\ 169 &= c^2\\ \sqrt{169} &= \sqrt{c^2}\\ 13 &= c\end{align*}

The length of the missing side is 13 centimeters.

Notice that as long as you use the Pythagorean Theorem you can figure out the missing length of any of the three sides of a right triangle.

III. Derive the Converse of the Pythagorean Theorem

If you use logic when thinking about the Pythagorean Theorem, there are many ways that you might find it useful. Always see how your knowledge might be applied to help you on a tough problem.

Example

Classify the triangle below as acute, right, or obtuse.

This triangle is specifically drawn not to scale. Therefore, you cannot decide whether the triangle is acute, right, or obtuse just by looking at it. Take a moment to analyze the side lengths and see how they are related. Two of the sides (15 and 17) are relatively close in length. The third side (8) is about half the length of the two longer sides.

To see if the triangle might be right, try plugging the values into the Pythagorean theorem to see if it makes it true. The hypotenuse is always the longest side, so 17 should be set equal to c\begin{align*}c\end{align*}. The other two values can represent a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*}.

a2+b282+152(8×8)+(15×15)64+225289=c2=172=(17×17)=289=289\begin{align*}a^2 + b^2 &= c^2\\ 8^2+15^2&=17^2\\ (8 \times 8) + (15 \times 15) &= (17 \times 17)\\ 64 + 225 &= 289\\ 289 &= 289\end{align*}

Since both sides of the equation are equal, the Pythagorean Theorem is true.

Therefore, the triangle described in the problem is a right triangle. We can use this logic to determine whether a triangle is a right triangle or not. Using this logic is referred to as using the converse of the Pythagorean Theorem. The Pythagorean theorem states that in a right triangle, a2+b2=c2\begin{align*}a^2 + b^2 = c^2\end{align*}. The converse of the Pythagorean theorem states that if a2+b2=c2\begin{align*}a^2 + b^2 = c^2\end{align*}, the triangle is a right triangle.

IV. Use the Converse of the Pythagorean Theorem to Identify Right Triangles

In the last example we derived the converse of the Pythagorean Theorem to figure out whether or not the triangle pictured was a right triangle or an acute triangle or an obtuse triangle. We can use the converse to prove whether or not a triangle is a right triangle. Remember, if the Pythagorean Theorem works for the values of the triangle, then the triangle is a right triangle. If not, then the triangle is not a right triangle.

Example

Identify whether the triangle below is a right triangle.

This triangle is specifically drawn not to scale. Therefore, you cannot decide whether the triangle is acute, right, or obtuse just by looking at it. Take a moment to analyze the side lengths and see how they are related. Two of the sides (5 and 8) are relatively close in length. The third side (12) is longer.

To see if the triangle might be right, try plugging the values into the Pythagorean theorem to see if it makes it true. The hypotenuse is always the longest side, so 12 should be set equal to c\begin{align*}c\end{align*}. The other two values can represent a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*}.

a2+b252+82(5×5)+(8×8)25+6489=c2=122=(12×12)=144144\begin{align*}a^2 + b^2 &= c^2\\ 5^2 + 8^2 &= 12^2\\ (5 \times 5) + (8 \times 8) &= (12 \times 12)\\ 25 + 64 &= 144\\ 89 &\ne 144\end{align*}

At the end of the solution, you can see that the result on the left side was 89, and the result on the right side was 144. Therefore, the sum of the squares of the legs did not equal the square of the hypotenuse. So, the triangle is not a right triangle.

Remember to use the Pythagorean Theorem whenever you want to prove that a triangle is or is not a right triangle. Now let’s use what we have learned on the problem from the introduction.

## Real-Life Example Completed

The Painting Project

Here is the problem from the introduction. Reread it and then explain the mathematical formula and why it is needed.

While Mr. Kennedy’s class was working on the garden, the students in Ms. Richardson’s class decided to paint the equipment shed where all of the sports equipment was kept.

“That thing hasn’t been painted in decades,” Karen said in the first meeting.

“I agree, it does look awful,” Cameron added.

“Well, I don’t know about decades, but it does need to be painted, so that is what we are going to do. Now to work on this project, we will need to choose a ladder that will reach up high enough. How can we figure this out?” Ms. Richardson asked.

The class was silent.

“I have an idea,” Veria said smiling. “It has to do with triangles. We need to figure out the height of the ladder, compared to the height of the building.”

“Yes, but don’t forget that the ladder reaches out from the building, not up against it. So we have to consider that measurement too,” Aran chimed in.

“What can we use to figure that out?” Karen asked.

Once again, the group was silent.

Solution to Real – Life Example

The students will need to use the Pythagorean Theorem to figure out this problem.

\begin{align*}a^2 + b^2 = c^2\end{align*}

Why? They will need to use the Pythagorean Theorem because the ladder against the shed forms a right triangle with the ground. The shed and the distance that the ladder is placed from the shed form the sides \begin{align*}a\end{align*} and \begin{align*}b\end{align*} of the right triangle. The ladder is the \begin{align*}c\end{align*} side of the triangle.

Look at the diagram below.

## Vocabulary

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

Right Triangle
one angle is equal to \begin{align*}90^{\circ}\end{align*}.
Legs
the two shorter sides of a right triangle.
Hypotenuse
the longest side of a right triangle.
Pythagorean Theorem
\begin{align*}a^2 + b^2 = c^2\end{align*}
Pythagorean Triple
values that work perfectly in the Pythagorean Theorem. The ratio always simplifies to 3:4:5.
Converse of the Pythagorean Theorem
If \begin{align*}a^2 + b^2 = c^2\end{align*}, then the triangle is a right triangle.

## Time to Practice

Directions: Use the Pythagorean Theorem to find the missing dimensions of right triangles.

1. \begin{align*}a=3,b=4,c=?\end{align*}
2. \begin{align*}a=6,b=8,c=?\end{align*}
3. \begin{align*}a=9,b=12,c= ?\end{align*}
4. \begin{align*}a=27,b=36,c= ?\end{align*}
5. \begin{align*}a=15,b=20,c= ?\end{align*}
6. \begin{align*}a=18,b=24,c= ?\end{align*}
7. \begin{align*}a= ?,b=16,c= 20\end{align*}
8. \begin{align*}a= ?,b=28,c=35\end{align*}
9. \begin{align*}a=30,b= ?,c=50\end{align*}
10. \begin{align*}a=33,b= ?,c=55\end{align*}
11. \begin{align*}a=1.5,b= ?,c=2.5\end{align*}
12. \begin{align*}a=36,b= ?,c=60\end{align*}

Directions: Think about what you have learned about the Pythagorean Theorem and answer true or false for the following questions.

1. The Pythagorean Theorem will work for an acute triangle with all \begin{align*}60^{\circ}\end{align*} angles.
2. The Pythagorean Theorem will work for a right triangle.
3. The Pythagorean Theorem will only work if the triangle is a right triangle.
4. The legs of a right triangle are considered the two shorter sides of the right triangle.
5. The hypotenuse is the longest side of a right triangle.
6. The converse of the Pythagorean Theorem is used to find the angle measures of an obtuse triangle.
7. A Pythagorean Triple is when you multiply all of the angle measures by three.
8. You can use the Pythagorean Theorem to figure out if the side lengths of a triangle make it a right triangle or not.

Directions: Identify whether or not each of the following values is a Pythagorean Triple. Write yes or no for your answer.

1. 4, 5, 6
2. 6, 8, 10
3. 5, 6, 9
4. 9, 12, 15
5. 30, 40, 55
6. 21, 28, 35
7. 12, 16, 20

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