Lesson 8 – 2 The Pythagorean Theorem and Its Converse Geometry Lesson 8 – 2 The Pythagorean Theorem and Its Converse Objective: Use the Pythagorean Theorem. Use the Converse of the Pythagorean Theorem.

Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

Find the missing measure.

Find x

Pythagorean Triple A set of nonzero whole numbers a, b, and c such that a2 + b2 = c2. Can you think of a Pythagorean triple? 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 6, 8, 10 10, 24, 26 16, 30, 34 14, 48, 50

Use a Pythagorean triple to find x. Explain. Reduce 15 & 12 by 3. What number completes the triple? 5 4 3, 4, 5 Multiply 3 by 3 (since we reduced by 3 earlier) So x = 9

Use a Pythagorean triple to find x. Explain. 12 5 5, 12, 13 The numbers can be reduced so that 5, 12, 13 is the triple So x = 13(4) which is 52

Real World 7 feet 11 feet 13 feet 17 feet Damon is locked out of his house, with the only open window on the second floor, 12 feet above the ground. He needs to borrow a ladder from his neighbor. If he must place the ladder 5 feet from the house to avoid some bushes, what length of ladder does Damon need? 7 feet 11 feet 13 feet 17 feet

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

Pythagorean Inequality Theorem 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 ides, then the triangle is an acute triangle.

Pythagorean Inequality Theorem 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, then the triangle is an obtuse triangle.

Since 2 sides is greater than third side, it is a triangle. Determine whether each set of numbers can be the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. 7, 14, 16 What kind of triangle? Is it a triangle? Review from lesson 5-5. 72 + 142 ? 162 7 + 14 > 16 7 + 16 > 14 14 + 16 > 7 245 256 < Since the square of the hypotenuse is larger: Since 2 sides is greater than third side, it is a triangle. Obtuse triangle

Determine whether each set of numbers can be the sides of a triangle Determine whether each set of numbers can be the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. 9, 40, 41 92 + 402 ? 412 9 + 40 > 41 9 + 41 > 40 40 + 41 > 9 1681 1681 = The triangle is a right triangle. So it is a triangle.

Determine whether each set of numbers can be the sides of a triangle Determine whether each set of numbers can be the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. 11, 60, 61 6.2, 13.8, 20 112 + 602 ? 612 11 + 60 > 61 11 + 61 > 60 60 + 61 > 11 3721 = 3721 Right triangle 44 < 45 Obtuse triangle 6.2 + 13.8 > 20 6.2 + 20 > 13.8 13.8 + 20 > 6.2 Not a triangle

Short cut

Homework Pg. 545 1 – 8 all, 10 – 38 EOE, 50, 54 – 74 EOE