10-4 The Pythagorean Theorem If you do not have a calculator, please get one from the back wall! The Chapter 10 test is a NON calculator test! Algebra 1 Glencoe McGraw-Hill Jo Ann Evans

This is Pythagoras, a Greek mathematician who lived from about 585-500 B.C. Although the Pythagorean theorem is named after him, there are indications that this theorem was in use in northern Africa before Pythagoras wrote of it.

Any triangle that has a right angle (90°) is called a right triangle Any triangle that has a right angle (90°) is called a right triangle. The two sides that meet to form the right angle are called the legs. The side across from the right angle is called the hypotenuse. (hypotenuse) (leg) c a b (leg) Hypotenuse, not Hippotenuse! It doesn’t matter which leg is “a” and which leg is “b”, but the hypotenuse “c” is always the side directly across from the right angle.

The Pythagorean Theorem is a statement that describes the relationship among the three sides of a right triangle. (hypotenuse) (leg)2 + (leg)2 = (hypotenuse)2 (leg) c a a2 + b2 = c2 b (leg) For any right triangle, the sum of the squares of the legs of the triangle is equal to the square of the hypotenuse.

If necessary, round answers to the nearest hundredth If necessary, round answers to the nearest hundredth. The Chapter 10 test is a NON calculator test!

(leg)2 + (leg)2 = (hypotenuse)2 a = 4 a2 + b2 = c2 In a right triangle, the lengths of the legs are 4 cm and 6 cm. Find the length of the hypotenuse. Draw a diagram. c (leg)2 + (leg)2 = (hypotenuse)2 a = 4 a2 + b2 = c2 Length will be a positive number, so find only the positive square root. b = 6 It doesn’t matter which leg is “a” and which leg is “b”, but the hypotenuse “c” is always the side directly across from the right angle. The length of the hypotenuse is cm or about 7.21 cm.

(leg)2 + (leg)2 = (hypotenuse)2 a = 13 In a right triangle, the length of a leg is 13 cm and the length of the hypotenuse is 17 cm. Find the length of the other leg. Draw a diagram. c = 17 (leg)2 + (leg)2 = (hypotenuse)2 a = 13 a2 + b2 = c2 b It doesn’t matter which leg is “a” and which leg is “b”, but the hypotenuse “c” is always the side directly across from the right angle. Length is cm or about 10.95 cm.

(leg)2 + (leg)2 = (hypotenuse)2 Example 1 In a right triangle, the lengths of the legs are 6 cm and 8 cm. Find the length of the hypotenuse. Draw a diagram. c = 10 a = 6 (leg)2 + (leg)2 = (hypotenuse)2 a2 + b2 = c2 b = 8 It doesn’t matter which leg is “a” and which leg is “b”, but the hypotenuse “c” is always the side directly across from the right angle. The length of the hypotenuse is 10 cm.

Example 2 In a right triangle, the length of a leg is 12 cm and the length of the hypotenuse is 15 cm. Find the length of the other leg. Draw a diagram. c = 15 (leg)2 + (leg)2 = (hypotenuse)2 a = 12 a2 + b2 = c2 b The length of the other leg is 9 cm.

The diagonal length of the TV screen is inches or about 22.36 inches. Example 3 Find the diagonal length of a TV screen that is 10 in wide by 20 inches long. a2 + b2 = c2 20 c 10 The diagonal length of the TV screen is inches or about 22.36 inches.

Example 4 Find the unknown length. Be careful! Remember to square the whole side. 10” a2 + b2 = c2 3x 8” If x = 2, then the unknown length is 3(2). The leg is 6” long.

Example 5 What is the longest line you can draw on a poster that is 15 inches by 25 inches? a2 + b2 = c2 25 15 The longest line possible would be about 29.15 inches long.

Example 6 Solve for x to find the missing lengths of the right triangle. a2 + b2 = c2 x The lengths of the triangle are 3, 4, and 5.

Example 7 A right triangle has one leg that is 2 inches longer than the other leg. The hypotenuse is 10 inches. Find the unknown lengths. 10 a2 + b2 = c2 x a c Substitute. b x + 2 Simplify. Write in standard form. Factor. Length is positive, so one length is 6 in and the other length is 8 in.

Example 8 A right triangle has one leg that is 3 inches longer than the other leg. The hypotenuse is 15 inches. Find the unknown lengths. 15 x a c a2 + b2 = c2 b x + 3 One leg is 9 in and the other leg is 12 in. –108 12 –9 3

Example 1 The length of the hypotenuse is 10 cm. Example 2 The length of the other leg is 9 cm. Example 3 The diagonal length is or about 22.36 in. Example 4 The leg is 6” long. Example 5 The longest line possible would be about 29.15 inches long. Example 6 The lengths of the triangle are 3, 4, and 5. Example 7 Length is positive, so one length is 6 in and the other length is 8 in. Example 8 One leg is 9 in and the other leg is 12 in.

Homework 10-A9 Page 552-554 #10-25, 53-58.