ESSENTIAL QUESTION: What kind of triangle is there when c2 is not equal to a2 + b2

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

Is ∆ABC a right triangle? Example 1 Verify a Right Triangle Is ∆ABC a right triangle? SOLUTION Compare c2 with a2 + b2. c2 a2 + b2 ? = Substitute 20 for c, 12 for a, and 16 for b. 202 122 + 162 ? = Multiply. 400 144 + 256 ? = 400 = 400 Simplify. ANSWER It is true that c2 = a2 + b2. So, ∆ABC is a right triangle. 3

c2 = a2 + b2 RIGHT c2 > a2 + b2 OBTUSE c2 < a2 + b2 ACUTE Right Triangle???? c2 = a2 + b2 RIGHT c2 > a2 + b2 OBTUSE c2 < a2 + b2 ACUTE

I DO…. Example 2 Show that the triangle is an acute triangle. SOLUTION Compare the side lengths. Compare c2 with a2 + b2. c2 a2 + b2 ? = Substitute for c, 4 for a, and 5 for b. 35 2 42 + 52 ? = Multiply. 35 16 + 25 ? = 35 < 41 Simplify. ANSWER Because c2 < a2 + b2, the triangle is acute. 5

I DO…. Example 3 Show that the triangle is an obtuse triangle. SOLUTION Compare the side lengths. c2 a2 + b2 ? = 225 64 + 144 Multiply. ? = 225 > 208 Simplify. ANSWER Because c2 > a2 + b2, the triangle is obtuse. 6

WE DO…. Example 4 Classify the triangle as acute, right, or obtuse. SOLUTION c2 a2 + b2 ? = 82 52 + 62 ? = 64 25 + 36 Multiply. ? = 64 > 61 Simplify. ANSWER Because c2 > a2 + b2, the triangle is obtuse. 7

Example 5 WE DO.… Classify the triangle with the given side lengths as acute, right, or obtuse. a. 4, 6, 7 b. 12, 35, 37 SOLUTION a. c2 a2 + b2 ? = b. c2 a2 + b2 ? = 72 42 + 62 ? = 372 122 + 352 ? = 49 16 + 36 ? = 1369 144 + 1225 ? = 49 < 52 1369 = 1369 The triangle is acute. The triangle is right. 8

Checkpoint YOU DO…. Classify the triangle as acute, right, or obtuse. Explain. 1. 2. 3.

Classify the triangle as acute, right, or obtuse. Explain. Checkpoint Classify Triangles Classify the triangle as acute, right, or obtuse. Explain. 1. ANSWER obtuse; 62 > 22 + 52 2. ANSWER right; 172 = 82 + 152 3. ANSWER acute; 72 < 72 + 72

Checkpoint Classify Triangles Use the side lengths to classify the triangle as acute, right, or obtuse. 4. 7, 24, 24 5. 7, 24, 25 6. 7, 24, 26

Checkpoint Classify Triangles Use the side lengths to classify the triangle as acute, right, or obtuse. 4. 7, 24, 24 ANSWER acute 5. 7, 24, 25 ANSWER right 6. 7, 24, 26 ANSWER obtuse

Hw Practice 4.5B