A14 APPENDIX Review Finding the Hypotenuse of a Right Triangle In a right triangle, one leg has length 4 and the other has length 3. What is the length of the hypotenuse? Solution EXAMPLE 1 Since the triangle is a right triangle, we use the Pythagorean Theorem with a 4 = and b 3 = to find the length c of the hypotenuse. From equation (1), c a b c c 4 3 16 9 25 25 5 2 2 2 2 2 2 = + = + = + = = = CONVERSE OF THE PYTHAGOREAN THEOREM In a triangle, if the square of the length of one side equals the sum of the squares of the lengths of the other two sides, the triangle is a right triangle.The 90° angle is opposite the longest side. Verifying That a Triangle Is a Right Triangle Show that a triangle whose sides are of lengths 5, 12, and 13 is a right triangle. Identify the hypotenuse. Solution EXAMPLE 2 Square the lengths of the sides. 5 25 12 144 13 169 2 2 2 = = = A.2 Geometry Essentials OBJECTIVES 1 Use the Pythagorean Theorem and Its Converse (p. A14) 2 Know Geometry Formulas (p. A15) 3 Understand Congruent Triangles and Similar Triangles (p. A16) 1 Use the Pythagorean Theorem and Its Converse The Pythagorean Theorem is a statement about right triangles . A right triangle is one that contains a right angle – that is, an angle of 90° . The side of the triangle opposite the 90° angle is called the hypotenuse ; the remaining two sides are called legs . In Figure 13 we have used c to represent the length of the hypotenuse and a and b to represent the lengths of the legs. Notice the use of the symbol to show the 90° angle. We now state the Pythagorean Theorem. 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. That is, in the right triangle shown in Figure 13, c a b 2 2 2 = + (1) Now Work problem 15 The converse of the Pythagorean Theorem is also true. Figure 13 A right triangle b Leg a Leg Hypotenuse c 905
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