9.2 The Pythagorean Theorem Unit IIB Day 5

Do Now (This is an SAT problem!)

Theorem 9.5: Converse of the Pythagorean Theorem If the square of the length of the longest side of the triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. If c2 = a2 + b2, then ∆ABC is a right triangle.

Ex. 1: Verifying Right Triangles These triangles appear to be right triangles. Tell whether they are right triangles or not. a. b. Let c represent the length of the longest side of the triangle. Check to see whether the side lengths satisfy the equation c2 = a2 + b2. (√113)2 = 72 + 82 113 = 49 + 64 113 = 113 ✔ c2 = a2 + b2. (4√95)2 = 152 + 362 42 ∙ (√95)2 = 152 + 362 16 ∙ 95 = 225+1296 1520 ≠ 1521 X

Ex. 1A These triangles appear to be right triangles. Tell whether they are right triangles. no yes

What if… c2 < a2 + b2? c2 > a2 + b2?

Theorem 9.6 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, then the triangle is acute. If c2 < a2 + b2, then ∆ABC is acute

Theorem 9.7 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 obtuse. If c2 > a2 + b2, then ∆ABC is obtuse

Triangle Inequality Theorem Any side of a triangle is always shorter than the sum of the other two sides. Converse: A triangle cannot be constructed from three segments if any of them is longer than the sum of the other two.

Ex. 2: Classifying Triangles Decide whether the set of numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute or obtuse. 38, 77, 86 10.5, 36.5, 37.5 You can use the Triangle Inequality to confirm that each set of numbers can represent the side lengths of a triangle. Compare the square of the length of the longest side with the sum of the squares of the two shorter sides. a) c2 ? a2 + b2 862 ? 382 + 772 7396 ? 1444 + 5959 7395 > 7373 The triangle is obtuse b) c2 ? a2 + b2 37.52 ? 10.52 + 36.52 1406.25 ? 110.25 + 1332.25 1406.24 < 1442.5 The triangle is acute.

Ex. 2A Decide whether the set of numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute, or obtuse. 8, 18, 24 3.2, 4.8, 5.1 yes; obtuse yes; acute

Closure Describe how to classify a triangle with side lengths 6, 9, and 10. Square all three sides. Since 102 < 62 + 92, the triangle is acute.