9.3 Converse of the Pythagorean Theorem Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Goals Determine if a triangle is a right triangle. Use the Pythagorean inequalities to determine if a triangle is acute or obtuse. February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. If ABC is a right triangle, then a2 + b2 = c2 a b c A B C February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Converse of 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. If a2 + b2 = c2, then ABC is a right triangle. a b c A B C February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Example 1 Is POD a right triangle? P O D 30 16 34 Longest Side Yes! February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Reminder February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Example 2 Which segment is the longest? Is HUG a right triangle? HG H U G 5 10 Yes! February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Example 3 Which segment is the longest? Is SAD a right triangle? SD S A D 9 12 20 No! February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Your Turn. Is RST a right ? S 24 10 Yes it is. R T 26 February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Triangle Inequality Theorem In a triangle, the sum of any two sides is greater than the third side. 4 + 7 > 5 4 + 5 > 7 5 + 7 > 4 7 4 5 February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Triangle Inequality Theorem This is not a triangle since 5 + 4 < 10. 4 5 10 February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Inequalities February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Begin with a right triangle… a2 + b2 = c2 c a b February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Rotate side a in. a and b have not changed. a2 + b2 has not changed. c got smaller. c2 got smaller. and… The right angle gets smaller: it is acute. c c a a b c2 < a2 + b2 c2 = a2 + b2 February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem If the square of the length of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is acute. A B C a b c c2 < a2 + b2 February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Take another right triangle… a2 + b2 = c2 c a b February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Rotate side a out. a and b have not changed. a2 + b2 has not changed. c got larger. c2 got larger. and… The right angle gets larger: it is obtuse. c a c a b c2 > a2 + b2 c2 = a2 + b2 February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem If the square of the length of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is obtuse. A B C a b c c2 > a2 + b2 February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Example 4 The sides of a triangle measure 5, 7, and 11. Classify it as acute, right, or obtuse. Solution: The longest side is 11. 112 ? 52 + 72 121 ? 25 + 49 121 > 74 Obtuse February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Example 5 The sides of a triangle are 17, 20, and 25. Classify the triangle. Solution: 252 ? 172 + 202 625 ? 689 625 < 689 Acute February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Example 6 Classify this triangle. ? ? Right  February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Example 7 Classify this triangle. It isn’t a triangle! 6 +8 < 16. 16 6 8 February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem

Geometry 9.3 Converse of the Pythagorean Theorem Summary If c2 = a2 + b2, RIGHT . If c2 < a2 + b2, ACUTE . If c2 > a2 + b2, OBTUSE . The last two can be very confusing; don’t get them mixed up. February 19, 2019 Geometry 9.3 Converse of the Pythagorean Theorem