AMSCO'S Geometry. New York - Rye High School

516 Ratio, Proportion, and SimilarityThe Converse of the Pythagorean TheoremIf we know the lengths of the three sides of a triangle, we can determine whetherthe triangle is a right triangle by using the converse of the Pythagorean Theorem.Theorem 12.17bIf the square of the length of one side of a triangle is equal to the sum ofthe squares of the lengths of the other two sides, then the triangle is a righttriangle.GivenABC with AB c, BC a, CA b, andc 2 a 2 b 2CFb a b aProveABC is a right triangle with C the rightangle.AcBDEProof Draw DEF with EF a, FD b, and F a right angle. Then DE 2 a 2 b 2 ,DE 2 c 2 and DE c. Therefore, ABC DEF by SSS. Correspondingangles of congruent triangles are congruent, so C F and C is a rightangle. Triangle ABC is a right triangle.We can state Theorems 12.17a and 12.17b as a single theorem.Theorem 12.17A triangle is a right triangle if and only if the square of the length of the longestside is equal to the sum of the squares of the lengths of the other two sides.EXAMPLE 1What is the length of the altitude to the base of an isosceles triangle if the lengthof the base is 18 centimeters and the length of a leg is 21 centimeters? Roundyour answer to the nearest centimeter.SolutionAnswerThe altitude to the base of an isosceles triangle is perpendicular to the base andbisects the base. In ABC, CD is the altitude to the base AB, AC is thehypotenuse of right triangle ACD, AD 9.0 cm, and AC 21 cm.AD 2 CD 2 AC 2C9 2 CD 2 21 221 cm81 CD 2 441CD 2 360CD "360 5 "36"10 5 6"10 19A 9.0 cm D BThe length of the altitude is approximately 19 centimeters.