7-2 The Pythagorean Theorem 3/2/17 7-2 The Pythagorean Theorem Objective: To use the Pythagorean Theorem and its converse. THEOREM 7-4 PYTHAGOREAN THEOREM In a right triangle, the sum of the squares of the length of the legs is equal to the square of the length of the hypotenuse a2 + b2 = c2 PYTHAGOREAN TRIPLE: a set of nonzero whole numbers a, b, and c that satisfy the equation a2 + b2 = c2 Examples of triples on next slide…. c a b

Are these Pyth. Triples?: 1) 20, 21, 29 2) 12, 16, 25 Ex: 3, 4, 5 5, 12, 13 7, 24, 25 8, 15, 17 Are these Pyth. Triples?: 1) 20, 21, 29 2) 12, 16, 25 Ex: Find x. Leave answer in simplified radical form. a2 + b2 = c2 Pythagorean Theorem 82 + x2 = 202 Substitution 20 64 + x2 = 400 Simplify 8 x2 = 336 Subtract x x = 336 Take Square Root x = 16•21 Simplify x = 4 21 400 + 441 = 841 yes! 144 + 256 ≠ 625 nooo!

Ex: Find the area of the triangle 12 m 12 m 102 + h2 = 122 h Find h, then use it to find Area Ex: Find the area of the triangle 12 m 12 m 102 + h2 = 122 h 100 + h2 = 144 h2 = 44 12 m 20 m h = 44 h h = 4•11 10 h = 2 11 A = ½ bh where b = 20, h = 2 11 = ½ (20)(2 11) = 20 11 m2 Since we knew all three sides already, Heron’s formula could have also been used.

THEOREM 7-5 CONVERSE OF THE PYTHAGOREAN If the square of the length of one side is equal to the sum of the squares of the length of the other two sides, then the triangle is a right triangle. Ex: Is this a right triangle? c2 = a2 + b2 85 852 = 132 + 842 13 7225 = 169 + 7056 84 7225 = 7225 YES Is this? 16, 48, 50 256 + 2304 ≠ 2500 Nooo!

THEOREM 7-6 If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, the triangle is obtuse. If c2 > a2 + b2, the triangle is obtuse! c a b

THEOREM 7-7 If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, the triangle is acute. If c2 < a2 + b2, the triangle is acute! a c Ex: Acute, obtuse, or right? b 6, 11, 14 142 = 62 + 112 196 = 36 + 121 196 > 157 OBTUSE since c2 > a2 + b2 12, 13, 15 152 = 122 + 132 225 = 144 + 169 225 < 313 ACUTE since c2 < a2 + b2

100 > 25 + 64 therefore it’s obtuse Find x. Answer in simplest radical form. 1) 9 x 12 2) 8 14 x 3) Find the area of the triangle. Leave answer in simplest radical form. 11 in 11 in 20 in 4) The lengths of the sides of a triangle are 5 cm, 8 cm, and 10 cm. Is it acute, right or obtuse? 81 + 144 = x2 225 = x2 x = 15 64 + x2 = 196 x2 = 132 x = 4•33 x = 2 33 100 + h2 = 121 h2 = 21 h = 21 A = ½(20)( 21) A = 10 21 100 > 25 + 64 therefore it’s obtuse

Assignment: Page 360 #1 – 9, 16 – 26