Lesson 9.4 Geometry

1
Lesson 9.4 Geometrys 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
A
Given Triangle ABC with right angle ACB
c
b
Prove
B
C
a
1. Given
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
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, (leg)2 (adjacent
leg)(hypotenuse).
5. a2 c2 cx
5. Distributive Property
6. b2 cx
6. In a right triangle with an altitude drawn to
the hypotenuse, (leg)2 (adjacent
leg)(hypotenuse).
7. a2 b2 c2 cx cx
7. Addition Property
8. Algebra
8. a2 b2 c2
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 gt c2, then the triangle is
acute a2 b2 c2, then the triangle is
right a2 b2 lt c2, then the triangle is obtuse
5
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
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
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
5
8
V
If 52 72 gt 82, then the triangle is acute If
52 72 82, then the triangle is right If 52
72 lt 82, then the triangle is obtuse
The triangle is acute
9
Summary State how to classify triangles. Explain
in your own words the Pythagorean Theorem.
Homework Worksheet 9.4