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)

Prove: Given: Triangle ABC with right angle ACB A BC b a c 1. ACB is a right Angle1. Given 2. From a point outside a line, only one perpendicular can be drawn to the line. 2. Draw CD to AB 3. A segment drawn from a vertex of a triangle perpendicular to the opposite side is an altitude. 3. CD is an altitude 4. a 2 = (c – x)c D x c – x 4. In a right triangle with an altitude drawn to the hypotenuse, (leg) 2 = (adjacent leg)(hypotenuse). 5. a 2 = c 2 – cx5. Distributive Property 6. b 2 = cx6. In a right triangle with an altitude drawn to the hypotenuse, (leg) 2 = (adjacent leg)(hypotenuse). 7. a 2 + b 2 = c 2 – cx + cx7. Addition Property 8. a 2 + b 2 = c 2 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 a 2 + b 2 > c 2, then the triangle is acute a 2 + b 2 = c 2, then the triangle is right a 2 + b 2 < c 2, then the triangle is obtuse

Example 1: Solve for x 8 6 x Use the Pythagorean Theorem = x = x = x 2 10 = x 10 = x Why do we not use -10? Example 2: Find the perimeter of the rectangle shown. x = x, P = 34 ( )

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. Since all sides are congruent, the perimeter is. x 5 3

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? Example 5: Find the altitude of an isosceles trapezoid whose sides have lengths of 10, 30, 10, and meters Altitude =

Example 6: Classify the triangle shown 7 58 ST V The triangle is acute If > 8 2, then the triangle is acute If = 8 2, then the triangle is right If < 8 2, then the triangle is obtuse

Summary State how to classify triangles. Explain in your own words the Pythagorean Theorem. Homework: Worksheet 9.4