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# Lengths of Triangle Sides Using the Pythagorean 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 $a^2 + b^2 = c^2$ where $a$ and $b$ are the legs of a right triangle and $c$ is the hypotenuse. Also, the side opposite the angle is lower case and the angle is upper case. For example, angle $A$ is opposite side $a$ .

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: $a = 8, \ b = 15$ , we need to find the hypotenuse.

$8^2 + 15^2 & = c^2\\64 + 225 & = c^2\\289 & = c^2\\17 & = c$

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.

$\left ( 5\sqrt{7} \right )^2 + x^2 & = \left ( 5\sqrt{13} \right )^2\\25 \cdot 7 + x^2 & = 25 \cdot 13\\175 + x^2 & = 325\\x^2 & = 150\\x & = 5\sqrt{6}$

#### Example C

Use the Pythagorean Theorem to find the missing side.

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

$10^2 + x^2 & = \left ( 10\sqrt{2} \right )^2\\100 + x^2 & = 100 \cdot 2\\100 + x^2 & = 100\\x^2 & = 100\\x & = 10$

### Vocabulary

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 $a$ , $b$ , and $c$ are lengths of the triangle.

### 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. $a = 1, \ b = 8$ , we need to find the hypotenuse.

$1^2 + 8^2 & = c^2\\1 + 64 & = c^2\\65 & = c^2\\\left ( \sqrt{65} \right ) & = c$

2. $a = 3, \ c = 11$ , we need to find the length of side $b$ .

$3^2 + b^2 & = 11^2\\9 + b^2 & = 11^2\\121 - 9 & = b^2\\112 = b^2\\\left ( \sqrt{112} \right ) & = b$

3. 2. $a = 7, \ c = 18$ , we need to find the length of side $b$ .

$7^2 + b^2 & = 18^2\\49 + b^2 & = 18^2\\324 - 49 & = b^2\\275 = b^2\\\left ( \sqrt{275} \right ) & = b$

### 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:

$a^2 + b^2 & = c^2\\9^2 + 12^2 & = c^2\\81 + 144 & = c^2\\225 = c^2\\\left ( \sqrt{225} \right ) & = 15 = c$

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.

### Practice

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 $x$ , 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 $2\sqrt{5}$ 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 $4\sqrt{7}$ and the hypotenuse is $10\sqrt{6}$ , then the other leg is _____________.
12. If the legs of a right triangle are $x$ and $y$ , 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 $(a + b)$ .
2. Find the sum of the areas of the square with sides $c$ and the right triangles with legs $a$ and $b$ .
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.