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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)
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