Lesson 9.4 Geometry’s Most Elegant Theorem

1. 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

2. 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)

3. The Picture Proof of the Pythagorean Theorem will be coming….. TOMORROW! Get excited….. It’s going to be awesome!

4. 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

5. 10 = x Example 1: Solve for x 8 Use the Pythagorean Theorem 6 x 82 + 62 = x2 64 + 36 = x2 100 = x2 10 = x Why do we not use -10? Example 2: Find the perimeter of the rectangle shown. • = x, • P = 34 (5 + 5 + 12 + 12) 13 x 5

6. 3 x 5 Since all sides are congruent, the perimeter is . 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.

7. Altitude = 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.

8. Example 6: Classify the triangle shown 7 S T 8 5 V If 52 + 72 > 82, 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

9. Homework: Worksheet 9.4