AMSCO'S Geometry. New York - Rye High School

Pythagorean Theorem 517EXAMPLE 2When a right circular cone is cut by a plane through the vertex and perpendicularto the base of the cone, the cut surface is an isosceles triangle. The length ofthe hypotenuse of the triangle is the slant height of the cone, the length one ofthe legs is the height of the cone, and the length of the other leg is the radiusof the base of the cone. If a cone has a height of 24 centimeters and the radiusof the base is 10 centimeters, what is the slant height of the cone?SolutionUse the Pythagorean Theorem:(h s) 2 (h c) 2 r 2(h s) 2 24 2 10 2(h s) 2 676h s 26 cm Answerh ch srPythagorean TriplesWhen three integers can be the lengths of the sides of a right triangle, this set ofnumbers is called a Pythagorean triple. The most common Pythagorean triple is3, 4, 5:3 2 4 2 5 2If we multiply each number of a Pythagorean triple by some positive integerx, then the new triple created is also a Pythagorean triple because it will satisfythe relation a 2 b 2 c 2 . For example:If {3, 4, 5} is a Pythagorean triple, then {3x,4x,5x} is also a Pythagorean triplefor a similar triangle where the ratio of similitude of the second triangle to thefirst triangle is x :1.Let x 2. Then {3x,4x,5x} {6, 8, 10} and 6 2 8 2 10 2 .Let x 3. Then {3x,4x,5x} {9, 12, 15}, and 9 2 12 2 15 2 .Let x 10. Then {3x,4x,5x} {30, 40, 50}, and 30 2 40 2 50 2 .Here are other examples of Pythagorean triples that occur frequently:{5, 12, 13} or, in general, {5x,12x,13x} where x is a positive integer.{8, 15, 17} or, in general, {8x,15x,17x} where x is a positive integer.The 45-45-Degree Right TriangleThe legs of an isosceles right triangle are congruent. The measure of each acuteangles of an isosceles right triangle is 45°. If two triangles are isosceles right trianglesthen they are similar by AA. An isosceles right triangle is called a 45-45-degree right triangle.