### Applications of the Pythagorean Theorem

#### Find the Height of an Isosceles Triangle

One way to use The Pythagorean Theorem is to find the height of an isosceles triangle (see Example 1).

#### Prove the Distance Formula

Another application of the Pythagorean Theorem is the Distance Formula. We will prove it here.

Let’s start with point and point . We will call the distance between and .

Draw the vertical and horizontal lengths to make a right triangle.

Now that we have a right triangle, we can use the Pythagorean Theorem to find the hypotenuse, .

**Distance Formula:** The distance between and is .

#### Classify a Triangle as Acute, Right, or Obtuse

We can extend the converse of the Pythagorean Theorem to determine if a triangle is an obtuse or acute triangle.

**Acute Triangles:** If the sum of the squares of the two shorter sides in a right triangle is ** greater** than the square of the longest side, then the triangle is

*acute.*For and , if , then the triangle is acute.

**Obtuse Triangles:** If the sum of the squares of the two shorter sides in a right triangle is ** less** than the square of the longest side, then the triangle is

*obtuse.*For and , if , then the triangle is obtuse.

What if you were given an equilateral triangle in which all the sides measured 4 inches? How could you use the Pythagorean Theorem to find the triangle's altitude?

### Examples

#### Example 1

What is the height of the isosceles triangle?

Draw the altitude from the vertex between the congruent sides, which will bisect the base.

#### Example 2

Find the distance between (1, 5) and (5, 2).

Make and . Plug into the distance formula.

Just like the lengths of the sides of a triangle, distances are always positive.

#### Example 3

Graph , and . Determine if is acute, obtuse, or right.

Use the distance formula to find the length of each side.

Plug these lengths into the Pythagorean Theorem.

is an obtuse triangle.

*For Examples 4 and 5, determine if the triangles are acute, right or obtuse.*

#### Example 4

Set the longest side to .

The triangle is obtuse.

#### Example 5

Set the longest side to .

A triangle with side lengths 5, 12, 13.

so this triangle is right.

### Review

Find the height of each isosceles triangle below. Simplify all radicals.

Find the length between each pair of points.

- (-1, 6) and (7, 2)
- (10, -3) and (-12, -6)
- (1, 3) and (-8, 16)
- What are the length and width of a 42” HDTV? Round your answer to the nearest tenth.
- Standard definition TVs have a length and width ratio of 4:3. What are the length and width of a 42” Standard definition TV? Round your answer to the nearest tenth.

Determine whether the following triangles are acute, right or obtuse.

- 7, 8, 9
- 14, 48, 50
- 5, 12, 15
- 13, 84, 85
- 20, 20, 24
- 35, 40, 51
- 39, 80, 89
- 20, 21, 38
- 48, 55, 76

Graph each set of points and determine whether is acute, right, or obtuse, using the distance formula.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.3.

### Resources