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Lesson 9.4 Geometry’s Most Elegant Theorem Objective: After studying this section, you will be able to use the Pythagorean Theorem and its converse Theorem The square of the measure of the hypotenuse of a right triangle is equal to the sum of the squares of the measures of the legs. (Pythagorean Theorem) Given: Triangle ABC with right angle ACB A D c b Prove: a 2 b2 c 2 C a B 1. ACB is a right Angle 1. Given 2. Draw CD to AB 2. From a point outside a line, only one perpendicular can be drawn to the line. 3. A segment drawn from a vertex of a triangle 3. CD is an altitude perpendicular to the opposite side is an altitude. 4. a2 = (c – x)c 4. In a right triangle with an altitude drawn to the hypotenuse, (leg)2 = (adjacent leg)(hypotenuse). 5. a2 = c2 – cx 6. b2 = cx 7. a2 + b2 = c2 – cx + cx 8. a2 + b2 = c2 5. Distributive Property 6. In a right triangle with an altitude drawn to the hypotenuse, (leg)2 = (adjacent leg)(hypotenuse). 7. Addition Property 8. Algebra Theorem If the square of the measure of one side of a triangle equals the sum of the squares of the measures of the other two sides, then the angle opposite the longest side is a right angle. (Converse of the Pythagorean Theorem) If c is the length of the longest side of a triangle, and a2 + b2 > c2, then the triangle is acute a2 + b2 = c2, then the triangle is right a2 + b2 < c2, then the triangle is obtuse Example 1: Solve for x 8 Use the Pythagorean Theorem 6 x 82 + 62 = x2 64 + 36 = x2 100 = x2 10 = x 10 = x Why do we not use -10? Example 2: Find the perimeter of the rectangle shown. 12 = x, P = 34 (5 + 5 + 12 + 12) 13 x 5 Example 3: Find the perimeter of a rhombus with diagonals of 6 and 10. Remember that the diagonals of a rhombus are perpendicular bisectors of each other. 3 x 5 Since all sides are congruent, the perimeter is 4 34 . Example 4: Nadia skipped 3 m north, 2 m east, 4 m north, 13 m east, and 1 m north. How far is Nadia from where she started? 17 meters Example 5: Find the altitude of an isosceles trapezoid whose sides have lengths of 10, 30, 10, and 20. Altitude = 5 3 Example 6: Classify the triangle shown 7 S 8 52 72 82, T 5 V If + > then the triangle is acute If 52 + 72 = 82, then the triangle is right If 52 + 72 < 82, then the triangle is obtuse The triangle is acute Summary State how to classify triangles. Explain in your own words the Pythagorean Theorem. Homework: Worksheet 9.4