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# 1.1: Lengths of Triangle Sides Using the Pythagorean Theorem

Difficulty Level: At Grade Created by: CK-12
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Practice Lengths of Triangle Sides Using the Pythagorean Theorem
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You've just signed up to be an architect's assistant in a new office downtown. You're asked to draw a scale model of a sculpture for a business plaza. The sculpture has a large triangular piece where one of the angles between the sides is ninety degrees. This type of triangle is called a ‘‘right triangle’’. The architect you're working for comes into the room and tells you that the sides of the triangle that form the right angle are 9 feet and 12 feet. Can you tell how long the third side is?

### Finding the Length of Triangle Sides Using Pythagorean Theorem

From Geometry, recall that the Pythagorean Theorem is 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 a right triangle and c\begin{align*}c\end{align*} is the hypotenuse. Also, the side opposite the angle is lower case and the angle is upper case. For example, angle A\begin{align*}A\end{align*} is opposite side a\begin{align*}a\end{align*}.

The Pythagorean Theorem is used to solve for the sides of a right triangle.

#### Using the Pythagorean Theorem

1.

a=8, b=15\begin{align*}a = 8, \ b = 15\end{align*}, we need to find the hypotenuse.

82+152=c264+225=c2289=c217=c\begin{align*}8^2 + 15^2 = c^2\\ 64 + 225 = c^2\\ 289 = c^2\\ 17 = c\end{align*}

Notice, we do not include -17 as a solution because a negative number cannot be a side of a triangle.

2.

Use the Pythagorean Theorem to find the missing leg.

(57)2+x2=(513)2257+x2=2513175+x2=325x2=150x=56\begin{align*}\left ( 5\sqrt{7} \right )^2 + x^2 = \left ( 5\sqrt{13} \right )^2\\ 25 \cdot 7 + x^2 = 25 \cdot 13\\ 175 + x^2 = 325\\ x^2 = 150\\ x = 5\sqrt{6}\end{align*}

3. Use the Pythagorean Theorem to find the missing leg in the triangle above.

102+x2=(102)2100+x2=1002100+x2=100x2=100x=10\begin{align*}10^2 + x^2 = \left ( 10\sqrt{2} \right )^2\\ 100 + x^2 = 100 \cdot 2\\ 100 + x^2 = 100\\ x^2 = 100\\ x = 10\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem asking you to draw a scale model of a sculpture for a business plaza.

With your knowledge of the Pythagorean Theorem, you can see that the triangle has sides with lengths 9 feet and 12 feet. You work to find the hypotenuse:

a2+b2=c292+122=c281+144=c2225=c2(225)=15=c\begin{align*}a^2 + b^2 = c^2\\ 9^2 + 12^2 = c^2\\ 81 + 144 = c^2\\ 225 = c^2\\ \left ( \sqrt{225} \right ) = 15 = c\end{align*}

With the knowledge that the length of the third side of the triangle is 15 feet, you are able to construct your scale model with ease.

#### Example 2

Use the Pythagorean Theorem to find the missing side of the following triangle:

a=1, b=8\begin{align*}a = 1, \ b = 8\end{align*}, we need to find the hypotenuse.

12+82=c21+64=c265=c265=c\begin{align*}1^2 + 8^2 = c^2\\ 1 + 64 = c^2\\ 65 = c^2\\ \sqrt{65} = c\end{align*}

#### Example 3

Use the Pythagorean Theorem to find the missing side of the following triangle:

a=3, b=11\begin{align*}a = 3, \ b = 11\end{align*}, we need to find the length of side c\begin{align*}c\end{align*}, the hypotenuse.

32+112=c29+121=c2130=c2130=c\begin{align*}3^2 + 11^2 = c^2\\ 9 + 121 = c^2\\ 130 = c^2\\ \sqrt{130} = c\end{align*}

#### Example 4

Find the missing side of the right triangle below. Leave the answer in simplest radical form.

a=7, c=18\begin{align*}a = 7, \ c = 18\end{align*}, we need to find the length of side b\begin{align*}b\end{align*}.

72+b2=18249+b2=18232449=b2275=b2275=b\begin{align*}7^2 + b^2 = 18^2\\ 49 + b^2 = 18^2\\ 324 - 49 = b^2\\ 275 = b^2\\ \sqrt{275} = b\end{align*}

### Review

Find the missing sides of the right triangles. Leave answers in simplest radical form.

1. If the legs of a right triangle are 3 and 4, then the hypotenuse is _____________.
2. If the legs of a right triangle are 6 and 8, then the hypotenuse is _____________.
3. If the legs of a right triangle are 5 and 12, then the hypotenuse is _____________.
4. If the sides of a square are length 6, then the diagonal is _____________.
5. If the sides of a square are 9, then the diagonal is _____________.
6. If the sides of a square are x\begin{align*}x\end{align*}, then the diagonal is _____________.
7. If the legs of a right triangle are 3 and 7, then the hypotenuse is _____________.
8. If the legs of a right triangle are 25\begin{align*}2\sqrt{5}\end{align*} and 6, then the hypotenuse is _____________.
9. If one leg of a right triangle is 4 and the hypotenuse is 8, then the other leg is _____________.
10. If one leg of a right triangle is 10 and the hypotenuse is 15, then the other leg is _____________.
11. If one leg of a right triangle is 47\begin{align*}4\sqrt{7}\end{align*} and the hypotenuse is 106\begin{align*}10\sqrt{6}\end{align*}, then the other leg is _____________.
12. If the legs of a right triangle are x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}, then the hypotenuse is ____________.

Pythagorean Theorem Proof

Use the picture below to answer the following questions.

1. Find the area of the square in the picture with sides (a+b)\begin{align*}(a + b)\end{align*}.
2. Find the sum of the areas of the square with sides c\begin{align*}c\end{align*} and the right triangles with legs a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*}.
3. Explain why the areas found in the previous two problems should be the same value. Then, set the expressions equal to each other and simplify to get the Pythagorean Theorem.

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### Vocabulary Language: English

Pythagorean Theorem

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by $a^2 + b^2 = c^2$, where $a$ and $b$ are legs of the triangle and $c$ is the hypotenuse of the triangle.

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