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? After completing this Concept, you'll be able to solve problems like this one.

### Watch This

CK-12 Foundation: Applications of the Pythagorean Theorem

James Sousa: Pythagorean Theorem and Its Converse

### Guidance

##### 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 A).

##### Prove the Distance Formula

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

Let’s start with point

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

##### 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

**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

#### Example A

What is the height of the isosceles triangle?

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

#### Example B

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

Make

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

#### Example C

Graph

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

Plug these lengths into the Pythagorean Theorem.

CK-12 Foundation: Applications of the Pythagorean Theorem

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### Guided Practice

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

1.

2.

3. A triangle with side lengths 5, 12, 13.

**Answers:**

Set the longest side equal to

1.

The triangle is acute.

2.

The triangle is obtuse.

3.

### Explore More

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

A(3,−5),B(−5,−8),C(−2,7) A(5,3),B(2,−7),C(−1,5) A(1,6),B(5,2),C(−2,3) A(−6,1),B(−4,−5),C(5,−2)

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 8.3.