1. Obj. 23 The Pythagorean Theorem The student is able to (I can): Use the Pythagorean Theorem to solve problems. Use Pythagorean inequalities to classify triangles.

2. The Pythagorean Theorem (a2 + b2 = c2) states the relationship between the sides of a right triangle. Although it was named for Pythagoras (circa 500 B.C.), this relationship was actually known to earlier people, including the Babylonians, Egyptians, and the Chinese. A Babylonian tablet from 1800 B.C. listing sides of right triangles. 3. The Pythagorean Theorem allows us to find an unknown side of a right triangle if we know the other two sides. Remember: theRemember: theRemember: theRemember: the hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c. x 12 13 4. The Pythagorean Theorem allows us to find an unknown side of a right triangle if we know the other two sides. Remember: theRemember: theRemember: theRemember: the hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c. x2 + 122 = 132 x 12 13 5. The Pythagorean Theorem allows us to find an unknown side of a right triangle if we know the other two sides. Remember: theRemember: theRemember: theRemember: the hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c. x2 + 122 = 132 x2 + 144 = 169 x 12 13 6. The Pythagorean Theorem allows us to find an unknown side of a right triangle if we know the other two sides. Remember: theRemember: theRemember: theRemember: the hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c. x2 + 122 = 132 x2 + 144 = 169 x2 = 25 x 12 13 7. The Pythagorean Theorem allows us to find an unknown side of a right triangle if we know the other two sides. Remember: theRemember: theRemember: theRemember: the hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c. x2 + 122 = 132 x2 + 144 = 169 x2 = 25 x = 5 x 12 13 8. Examples Find the value of x. Reduce radicals to simplest form. 1. 2. 2 6 x x x-2 4 9. Examples Find the value of x. Reduce radicals to simplest form. 1. 2. 2 2 2 2 6 x+ =2 6 x x x-2 4 10. Examples Find the value of x. Reduce radicals to simplest form. 1. 2. 2 2 2 2 6 x+ = 2 4 36 x+ = 2 6 x x x-2 4 11. Examples Find the value of x. Reduce radicals to simplest form. 1. 2. 2 2 2 2 6 x+ = 2 4 36 x+ = 2 40 x= 2 6 x x x-2 4 12. Examples Find the value of x. Reduce radicals to simplest form. 1. 2. 2 2 2 2 6 x+ = 2 4 36 x+ = 2 40 x= x 2 10= 2 6 x x x-2 4 13. Examples Find the value of x. Reduce radicals to simplest form. 1. 2. 2 2 2 2 6 x+ = 2 4 36 x+ = 2 40 x= x 2 10= 2 2 2 4 (x 2) x+ = 2 6 x x x-2 4 14. Examples Find the value of x. Reduce radicals to simplest form. 1. 2. 2 2 2 2 6 x+ = 2 4 36 x+ = 2 40 x= x 2 10= 2 2 2 4 (x 2) x+ =xxxx ----2222 xxxx x2 -2x ----2222 -2x 4 2 6 x x x-2 4 15. Examples Find the value of x. Reduce radicals to simplest form. 1. 2. 2 2 2 2 6 x+ = 2 4 36 x+ = 2 40 x= x 2 10= 2 2 2 4 (x 2) x+ =xxxx ----2222 xxxx x2 -2x ----2222 -2x 4 2 2 16 x 4x 4 x+ + = 2 6 x x x-2 4 16. Examples Find the value of x. Reduce radicals to simplest form. 1. 2. 2 2 2 2 6 x+ = 2 4 36 x+ = 2 40 x= x 2 10= 2 2 2 4 (x 2) x+ =xxxx ----2222 xxxx x2 -2x ----2222 -2x 4 2 2 16 x 4x 4 x+ + = 2 6 x x x-2 4 17. Examples Find the value of x. Reduce radicals to simplest form. 1. 2. 2 2 2 2 6 x+ = 2 4 36 x+ = 2 40 x= x 2 10= 2 2 2 4 (x 2) x+ =xxxx ----2222 xxxx x2 -2x ----2222 -2x 4 2 2 16 x 4x 4 x+ + = 20 4x = 0 2 6 x x x-2 4 18. Examples Find the value of x. Reduce radicals to simplest form. 1. 2. 2 2 2 2 6 x+ = 2 4 36 x+ = 2 40 x= x 2 10= 2 2 2 4 (x 2) x+ =xxxx ----2222 xxxx x2 -2x ----2222 -2x 4 2 2 16 x 4x 4 x+ + = 20 4x = 0 20 = 4x x = 5 2 6 x x x-2 4 19. Pythagorean Triple A set of nonzero whole numbers a, b, and c, such that a2 + b2 = c2. Memorize these! Note: 3, 4, 5 is the onlyonlyonlyonly triple that contains three consecutive numbers. Pythagorean TriplesPythagorean TriplesPythagorean TriplesPythagorean Triples BaseBaseBaseBase 3, 4, 5 5, 12, 13 7, 24, 25 8, 15, 17 x2x2x2x2 6, 8, 10 10, 24, 26 14, 48, 50 16, 30, 34 x3x3x3x3 9, 12, 15 x4x4x4x4 12, 16, 20 x5x5x5x5 15, 20, 25 20. Examples Find the missing side of the right triangle. 1. 3, 4, ____ 2. 9, ____, 15 3. ____, 12, 13 4. 8, 15, ____ 21. Examples Find the missing side of the right triangle. 1. 3, 4, ____ 2. 9, ____, 15 3. ____, 12, 13 4. 8, 15, ____ 5555 22. Examples Find the missing side of the right triangle. 1. 3, 4, ____ 2. 9, ____, 15 3. ____, 12, 13 4. 8, 15, ____ 5555 12121212 23. Examples Find the missing side of the right triangle. 1. 3, 4, ____ 2. 9, ____, 15 3. ____, 12, 13 4. 8, 15, ____ 5555 12121212 5555 24. Examples Find the missing side of the right triangle. 1. 3, 4, ____ 2. 9, ____, 15 3. ____, 12, 13 4. 8, 15, ____ 5555 12121212 5555 17171717 25. Thm 5-7-1 Thm 5-7-2 Converse of the Pythagorean Theorem If a2 + b2 = c2, then the triangle is a right triangle. Pythagorean Inequalities Theorem If then the triangle is an obtuseobtuseobtuseobtuse triangle. If then the triangle is an acuteacuteacuteacute triangle. 2 2 2 c a b ,> + 2 2 2 c a b ,< + 26. Classifying Triangles right triangle obtuse triangle acute triangle > +2 2 2 c a b < +2 2 2 c a b= +2 2 2 c a b a b c a b c a b c 27. Examples Classify the following triangle measures as right, obtuse, or acute. 1. 5, 7, 10 102 = 100, 52 + 72 = 74 ObtuseObtuseObtuseObtuse 2. 16, 30, 34 342 = 1156, 162 + 302 = 1156 RightRightRightRight 3. 3.8, 4.1, 5.2 5.22 = 27.04, 3.82 + 4.12 = 31.25 AcuteAcuteAcuteAcute