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### Intro to Pythagorean Theorem and its Converse

#### Pythagorean Theorem

If a triangle has one 90˚ angle, how do you find the length of the sides? You use the Pythagorean Theorem!

**Remember:** the Pythagorean Theorem *only* works for *right triangles*!

**Legs:** The two segments forming the right angle.

**Hypotenuse:** the segment opposite the right angle.

\begin{align*}(\text{leg}_1)^2+(\text{leg}_2)^2 = (\text{hypotenuse})^2\end{align*}

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

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If you want to find one side, you plug in the other side lengths and solve.

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

**Find the missing sides:**

#### Converse of Pythagorean Theorem

If the lengths of three sides of a triangle make the equation \begin{align*}a^2+b^2=c^2\end{align*} true, then they represent the sides of a right triangle.

A.

**Are the following triangles right triangles? How do you know?**

The first two numbers are legs, the third is the possible hypotenuse.

- 3, 4, 5
- 6, 7, 10
- 5, 12, 13
- 7, 24, 25
- 4, 8, 12

**Hint:** Plug the numbers into the Pythagorean Theorem with the third number being "c". If it both sides of the equation are equal, the triangle is a right triangle.

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### Pythagorean Theorem Applications - Distance Formula

How can we use the Pythagorean Theorem to find the distance between two points on a graph?

*Think of the segment as the hypotenuse of a right triangle.* You can easily count the lengths of the two legs, then *use the Pythagorean Theorem* to find the length of the hypotenuse.

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Find the distance between the points (1,5) and (5,2):

*Hint: The legs are shown by these lines:*

Click here for the answer.

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**Find the distance between the two points:**

- (7, 7) and (–7, 7)
- (–3, 6) and (3, –6)
- (–3, –1) and (–5, –8)
- (3, –4) and (6, 0)
- (–1, 0) and (4, 2)
- (–3, 2) and (6, 2)

*Hint: Graph each point and use the Pythagorean Theorem to solve.*

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What are some other ways to use the Pythagorean Theorem?