Pythagorean Theorem Pre-Algebra

Pythagoras

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 Hypotenuse Leg

Pythagorean Theorem c b a 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. c b a

Finding the Hypotenuse a2 + b2 = c2 Example 1: Find the length of the hypotenuse of a right triangle if a = 3 and b = 4. 4 3 c

Finding the Length of a Leg Example 2: Find the length of the leg of the following right triangle. a2 + b2 = c2 12 __________________ a 9

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 13 c 11 a 4 5

Solution of Example 3 a2 + b2 = c2 c 11 4 Find the length of the hypotenuse c when a = 11 and b = 4. a2 + b2 = c2 c 11 4

Solution of Example 4 Example 4: Find the length of the leg of the following right triangle. _______________ 13 a 5

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. 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. This is a right triangle. A set of integers such as 12, 16, and 20 is a Pythagorean triple.

Converse Examples Example 7: Determine whether 4, 5, 6 is a Pythagorean triple. Example 8: Determine whether 15, 8, and 17 is a Pythagorean triple. 4, 5, and 6 is not a Pythagorean triple. 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 For the baseball diamond, a = 90 and b = 90. c 90 The distance from home plate to second base is approximately 127 feet. 90