Pythagorean Theorem

Pre-Algebra

ALCOS 7

Lesson Topics

• Baseball• Definitions• Pythagorean Theore

m• Converse of the

Pythagorean Theorem

• Application of the Pythagorean Theorem

Baseball A baseball scout uses many

different tests to determine whether or not to draft a particular player. One test for catchers is to see how quickly they can throw a ball from home plate to second base. The scout must know the distance between the two bases in case a player cannot be tested on a baseball diamond. This distance can be found by separating the baseball diamond into two right triangles.

Right Triangles

• Right Triangle – A triangle with one right angle.

• Hypotenuse – Side opposite the right angle and longest side of a right triangle.

• Leg – Either of the two sides that form the right angle.

Leg

Leg

Hypotenuse

Pythagorean Theorem

• In a right triangle, if a and b are the measures of the legs and c is the measure of the hypotenuse, then

a2 + b2 = c2.• This theorem is used to

find the length of any right triangle when the lengths of the other two sides are known.

b

a

c

Finding the Hypotenuse

• Example 1: Find the length of the hypotenuse of a right triangle if a = 3 and

b = 4.4

3c

a2 + b2 = c2

5

5

25

25

169

43

2

2

222

c

c

c

c

c

c

Finding the Length of a Leg

• Example 2: Find the length of the leg of the following right triangle.

9

12

a

a2 + b2 = c2

14481

14481

129

2

2

222

a

a

a

81 81__________________

94.7

63

632

a

a

a

Examples of the Pythagorean Theorem

• Example 3: Find the length of the hypotenuse c when a = 11 and b = 4. Solution

• Example 4: Find the length of the leg of the following right triangle.

Solution

11

4

c

5

13a

Solution of Example 3

• Find the length of the hypotenuse c when

a = 11 and b = 4.

a2 + b2 = c2

11

4

c

70.11

137

137

16121

411

2

2

222

c

c

c

c

c

Solution of Example 4

16925

1352

222

222

a

a

cba • Example 4: Find the length of the leg of the following right triangle.

13a

5

2525_______________

12

144

1442

a

a

a

Converse of the Pythagorean Theorem

• If a2 + b2 = c2, then the triangle with sides a, b, and c is a right triangle.

• If a, b, and c satisfy the equation

a2 + b2 = c2, then a, b, and c are known as Pythagorean triples.

Example of the Converse

Example 5: Determine whether a triangle with lengths 7, 11, and 12 form a right triangle.

**The hypotenuse is the longest length.

14412149

12117?

2?

22

144170

This is not a right triangle.

Example of the Converse

Example 6: Determine whether a triangle with lengths 12, 16, and 20 form a right triangle.

400256144

201612?

2?

22

400400 This is a right triangle. A set of integers such

as 12, 16, and 20 is a Pythagorean triple.

Converse ExamplesExample 7: Determine

whether 4, 5, 6 is a Pythagorean triple.

Example 8: Determine whether 15, 8, and 17 is a Pythagorean triple.

362516

654?

2?

22

36414, 5, and 6 is not a Pythagorean triple.

28964225

17815?

2?

22

289289 15, 8, and 17 is a Pythagorean triple.

Baseball Problem

• On a baseball diamond, the hypotenuse is the length from home plate to second plate. The distance from one base to the next is 90 feet. The Pythagorean theorem can be used to find the distance between home plate to second base.

Solution to Baseball Problem

2

222

222

81008100

9090

c

c

cba

• For the baseball diamond, a = 90 and

b = 90.

90

90

2200,16 cc200,16

127cThe distance from home plate to second base is

approximately 127 feet.

c