Pythagorean Theorem Pythagorean Theorem and Its Converseand Its Converse
ObjectiveObjective
To use the Pythagorean Theorem To use the Pythagorean Theorem and its converseand its converse
Essential Understanding:If you know the lengths of any two sides of a right
triangle, you can find the length of the third side by using the Pythagorean Theorem.
Pythagorean TheoremIn a right triangle, the sum of the square of the lengths of the legs equals the square of the length of the hypotenuse.
If a and b are the lengths of the legs, and c is the length of the hypotenuse, then a2 + b2 = c2.
a
b
c
Theorem: A conjecture that has been proved
72b
b
11
7
c8
6
How high up a wall will a 20 foot ladder touch if the foot of the ladder is placed 5 feet from the wall? Find the approximate height
a2 + b2 = c2
62 + 82 = c2
36 + 64 = c2
100 = c2
a2 + b2 = c2
72 + b2 = 112
49 + b2 = 121
b2 = 72
10 = c
26
b20 ft
a2 + b2 = c2
52 + b2 = 202
25 + b2 = 400b2 = 375
375b
5 ft
ft 4.19
Converse of Pythagorean Converse of Pythagorean TheoremTheorem
If the sum of the squares of the If the sum of the squares of the lengths of two sides of a triangle lengths of two sides of a triangle is equal to the square of the is equal to the square of the length of the third side, then the length of the third side, then the triangle is a right triangle.triangle is a right triangle.
Pythagorean TheoremGiven a right triangle then a
b
c
a2 + b2 = c2
Converse: Switch the if and then parts
Given 3 sides of triangle that satisfy a2 + b2 = c2
then triangle is a right triangle
24
26
10
87
4
a2 + b2 = c2
42 + 72 = 82
16 + 49 = 64
65 = 64
a2 + b2 = c2
102 + 242 = 262
100 + 576 = 676
676 = 676
No
Right Triangle?
Yes
The following theorems allow you to determine whether a triangle is acute or obtuse. These theorems relate to the Hinge Theorem, which states that the longer side is opposite the larger angle and the shorter side is opposite the smaller angle.
AND….
A triangle has side lengths 6, 11 and 14. Is it acute, obtuse or right?
p. 495: 7, 9, 11, 16-32 even
Leave answers in simplest radical form
