Lesson 9.4 Geometry`s Most Elegant Theorem

```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,
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,
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
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