Pythagorean Theorem and Its Converse

1. Pythagorean Theorem and Its Converse Objective To use the Pythagorean Theorem and its converse Essential Understanding: If you know the lengths of any two sides of a right triangle, you can find the length of the third side by using the Pythagorean Theorem.

2. Theorem: A conjecture that has been proved Pythagorean Theorem In a right triangle, the sum of the square of the lengths of the legs equals the square of the length of the hypotenuse. If a and b are the lengths of the legs, and c is the length of the hypotenuse, then a2 + b2 = c2. c a b

3. Examples How high up a wall will a 20 foot ladder touch if the foot of the ladder is placed 5 feet from the wall? Find the approximate height 7 b c 8 11 20 ft b a2 + b2 = c2 6 72 + b2 = 112 a2 + b2 = c2 5 ft a2 + b2 = c2 49 + b2 = 121 62 + 82 = c2 52 + b2 = 202 b2 = 72 36 + 64 = c2 25 + b2 = 400 b2 = 375 100 = c2 10 = c

4. Converse of Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.

5. Pythagorean Theorem Given a right triangle then a c a2 + b2 = c2 Converse of Pythagorean Theorem b Converse: Switch the if and then parts Given 3 sides of triangle that satisfy a2 + b2 = c2 then triangle is a right triangle

6. Examples Right Triangle? 10 24 8 7 26 a2 + b2 = c2 4 102 + 242 = 262 a2 + b2 = c2 100 + 576 = 676 42 + 72 = 82 676 = 676 16 + 49 = 64 65 = 64 Yes No

7. The following theorems allow you to determine whether a triangle is acute or obtuse. These theorems relate to the Hinge Theorem, which states that the longer side is opposite the larger angle and the shorter side is opposite the smaller angle.

8. AND….

9. A triangle has side lengths 6, 11 and 14. Is it acute, obtuseor right? p. 495: 7, 9, 11, 16-32 even Leave answers in simplest radical form