Lesson 9.4 Geometry’s Most Elegant Theorem Objective: After studying this section, you will be...

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)

2 2 2a b c

The Picture Proof of the Pythagorean Theorem will be

coming…..TOMORROW!

Get excited…..It’s going to be awesome!

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 acutea2 + b2 = c2, then the triangle is righta2 + b2 < c2, then the triangle is obtuse

Example 1: Solve for x8

6x

Use the Pythagorean Theorem

82 + 62 = x2

64 + 36 = x2

100 = x2

10 = x10 = x Why do we not use -10?

Example 2:

Find the perimeter of the rectangle shown.

x 13

5

12 = x,

P = 34 (5 + 5 + 12 + 12)

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 .4 34

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 20.

17 meters

Altitude = 5 3

Example 6:

Classify the triangle shown 7

58

S T

V

The triangle is acute

If 52 + 72 > 82, then the triangle is acuteIf 52 + 72 = 82, then the triangle is rightIf 52 + 72 < 82, then the triangle is obtuse

Homework:

Worksheet 9.4