# Distance Formula

## Using the Pythagorean Theorem to determine distances

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Pythagorean Theorem and the Distance Formula

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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.

(leg1)2+(leg2)2=(hypotenuse)2\begin{align*}(\text{leg}_1)^2+(\text{leg}_2)^2 = (\text{hypotenuse})^2\end{align*}

a2+b2=c2\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:

[Figure1]

[Figure2]

[Figure3]

#### 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:

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Find the distance between the two points:

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

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