Pythagorean Triples and Inequality Theorem Students will be

Pythagorean Triples and Inequality Theorem Students will be able to identify a Pythagorean triple and classify a triangle using the Pythagorean Inequality Theorem

Pythagorean Triples § A set of three nonzero whole numbers a, b and c such that a 2 + b 2 = c 2 is called a Pythagorean triple. § These are some common Pythagorean triples: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 § How can we identify a Pythagorean triple? § Use the Pythagorean Theorem to make sure all three numbers are whole numbers FHS Unit E 2

Identifying Pythagorean Triples Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. a 2+ b 2 = c 2 142 + 482 = c 2 2500 = c 2 50 = c The side lengths are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2, so they form a Pythagorean triple. FHS Unit E 3

Pythagorean Inequalities Theorem In ABC, c is the length of the longest side. • If > + then ΔABC is an obtuse triangle. c 2 FHS a 2 b 2, • If c 2 < a 2 + b 2, then ΔABC is an acute triangle. Unit E 4

Example of Classifying Triangles Tell if the measures 5, 7 and 10 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. Step 1: Determine if the measures form a triangle. By the Triangle Inequality Theorem, 5, 7, and 10 can be the side lengths of a triangle. Step 2: Classify the triangle. ? c =a + 2 2 ? 10 = 5 2 + 7 2 2 ? 100 = 25 + 49 b 2 100 > 74 Since c 2 > a 2 + b 2, the triangle is obtuse. FHS Unit E 5

Lesson Quiz 1. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 13; yes; the side lengths are nonzero whole numbers that satisfy Pythagorean’s Theorem. 2. Tell if the measures 9, 13, and 15 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. yes; acute 3. FHS Unit E 6