The Converse Of The Pythagorean Theorem Section 8-1

Objectives: Use the Converse of the Pythagorean Theorem. Use side lengths to classify triangles by their angles.

Converse of the Pythagorean Theorem 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 it is a right triangle. Example:

Example 1 Is ∆ABC a right triangle? SOLUTION 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. 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. 4

Example 2a: Determine whether 9, 12, and 15 are the sides of a right triangle. Since the measure of the longest side is 15, 15 must be c. Let a and b be 9 and 12. Pythagorean Theorem Simplify. Add. Answer: These segments form the sides of a right triangle since they satisfy the Pythagorean Theorem.

Example 2b: Determine whether 21, 42, and 54 are the sides of a right triangle. Pythagorean Theorem Simplify. Add. Answer: Since , segments with these measures cannot form a right triangle.

Example 2c: Determine whether 4√3, 4, and 8 are the sides of a right triangle. Pythagorean Theorem Simplify. Add. Answer: Since 64 = 64, segments with these measures form a right triangle.

Your Turn: Determine whether each set of measures are the sides of a right triangle. a. 6, 8, 10 b. 5, 8, 9 c. Answer: The segments form the sides of a right triangle. Answer: The segments do not form the sides of a right triangle. Answer: The segments form the sides of a right triangle.

More Practice Which of the following is a right triangle? NO NO 202(400)≠152+122(369) 272(729)≠202+152(625) 652(4225)=602+252(4225) NO YES 302(900)≠182+252(949)

Classifying triangles

Classifying Triangles Using the Converse of the Pythagorean Theorem we can classify a triangle as acute, right, or obtuse by its side lengths. Obtuse 1 angle is obtuse (measure > 90°) Right 1 angle is right (measure = 90°) Acute all 3 angles are acute (measure < 90°)

Classifying Triangles Acute Triangle 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 it is an acute triangle. Example:

Classifying Triangles Right Triangle 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 it is a right triangle. Example: Then triangle ABC is right

Classifying Triangles Obtuse Triangle 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 it is an obtuse triangle. Example:

Summary This is the Converse of the Pythagorean Theorem

Example 3 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. 16

Example 4 Show that the triangle is an obtuse triangle. SOLUTION Compare the side lengths. c2 a2 + b2 Compare c2 with a2 + b2. ? = (15)2 82 + 122 Substitute 15 for c, 8 for a, and 12 for b. ? = 225 64 + 144 Multiply. ? = 225 > 208 Simplify. ANSWER Because c2 > a2 + b2, the triangle is obtuse. 17

Example 5 Classify the triangle as acute, right, or obtuse. SOLUTION Compare the square of the length of the longest side with the sum of the squares of the lengths of the two shorter sides. c2 a2 + b2 Compare c2 with a2 + b2. ? = 82 52 + 62 Substitute 8 for c, 5 for a, and 6 for b. ? = 64 25 + 36 Multiply. ? = 64 > 71 Simplify. ANSWER Because c2 > a2 + b2, the triangle is obtuse. 18

Example 6 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 ? = 1369 144 + 1225 ? = 49 16 + 36 ? = 49 < 52 1369 = 1369 The triangle is acute. The triangle is right. 19

Your Turn: Classify the triangle as acute, right, or obtuse. Explain. 1. ANSWER obtuse; 62 > 22 + 52 36 >29 2. ANSWER right; 172 = 82 + 152 289 = 289 3. ANSWER acute; 72 < 72 + 72 49 < 98

Your Turn: 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

Example 7 Classify the triangle with lengths 9, 12, and 15 as acute, right, or obtuse. Justify your answer. c2 = a2 + b2 Compare c2 and a2 + b2. ? 152 = 122 + 92 Substitution ? 225 = 225 Simplify and compare. Answer: Since c2 = a2 + b2, the triangle is right.

Example 8 Classify the triangle with lengths 10, 11, and 13 as acute, right, or obtuse. Justify your answer. c2 = a2 + b2 Compare c2 and a2 + b2. ? 132 = 112 + 102 Substitution ? 169 < 221 Simplify and compare. Answer: Since c2 < a2 + b2, the triangle is acute.

Your Turn: A. Classify the triangle with lengths 7, 8, and 14 as acute, right, or obtuse. A. acute B. obtuse C. right

Your Turn: B. Classify the triangle with lengths 26, 22, and 33 as acute, right, or obtuse. A. acute B. obtuse C. right

Joke Time What is black, white and red all over? You fold it! A newspaper. How can you tell if there is an elephant in your fridge? You can't close the door. What did one wall say to the other? "I'll meet you in the corner."