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# Lengths of Triangle Sides Using the Pythagorean Theorem

## Discover, Geometrically prove, and apply the theorem

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Lengths of Triangle Sides Using the Pythagorean Theorem

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? When you've completed this Concept, you'll be able to find the length of an unknown side of a right triangle by using the lengths of the other two sides.

### Guidance

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.

#### Example A

Use the Pythagorean Theorem to find the missing side.

Solution: 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

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

#### Example B

Use the Pythagorean Theorem to find the missing side.

Solution: Use the Pythagorean Theorem to find the missing leg.

(57)2+x2=(513)2257+x2=2513175+x2=325x2=150x=56

#### Example C

Use the Pythagorean Theorem to find the missing side.

Solution: Use the Pythagorean Theorem to find the missing leg.

102+x2=(102)2100+x2=1002100+x2=100x2=100x=10

### Guided Practice

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

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

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

Solutions:

1. 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

2. 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

3. 2. 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

### Concept Problem Solution

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

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.

### Explore More

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.

### Vocabulary Language: English

Pythagorean Theorem

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.