Download Pythagorean Theorem and Triangles

Rules of Pythagoras
leg 2 (b)
All Triangles:
•The sum of the three angles
equals 180°.
leg 1 (a)
Right Triangles:
• One of its angles must be a right
angle (90°).
•The side opposite of the right
angle is called the hypotenuse (c).
•The other sides of the triangle are
called the legs (a and b).
Pythagorean Theorem
c
a
b
If it is a right triangle:
a2 + b2 = c2
Pythagorean Theorem
and its Converse
c
a
b
• If it is a right triangle then a2 + b2 = c2
• The converse states that if a2 + b2 = c2 then it must
be a right triangle
Right
Triangle
• Prove that the following
triangle is a right triangle
8
Be Careful! Assign c correctly!
1.
2.
3.
4.
17
15
If a2 + b2 = c2 then it is a right triangle
82 + 152 = 172
64 + 225 = 289
289 = 289 , it is a right triangle
Acute
Triangle
c
a
b
• A triangle with three acute (less than 90°) angles
• In non-right triangles a2 + b2 will not equal c2
• Still use the Pythagorean Theorem to determine if the
triangle is acute or obtuse
• If c2 < a2 + b2 then the triangle is acute
Acute
Triangle
• Prove that the following
triangle is an acute triangle
1
0
7
8
1.
2.
3.
4.
5.
If it was a right triangle then a2 + b2 = c2
72 + 82 ≠ 102
49 + 64 ≠ 100
113 > 100 or a2 + b2 > c2
Since 113 > 100 it is an acute triangle
Obtuse
Triangle
a
c
b
• A triangle with one obtuse (greater than 90°) angle
• In non-right triangles a2 + b2 will not equal c2
• Still use the Pythagorean Theorem to determine if the
triangle is acute or obtuse
• If c2 > a2 + b2 then the triangle is obtuse
Obtuse
Triangle
• Prove that the following
triangle is an obtuse triangle
10
6
7
1.
2.
If it was a right triangle then a2 + b2 = c2
62 + 72 ≠ 102
3.
4.
5.
36 + 49 ≠ 100
100 > 85 or c2 > a2 + b2
Since 100 > 85 it is an obtuse triangle
Practice
Find x:
Practice
If c is the measure of the hypotenuse, find each missing side:
1. a = 12, b = 9, c = ?
c = 15
2. a = 8, b = ?, c = 21
b = 19.4
Practice
Find the missing measure in each right triangle:
1.
2.
c = 12.6
x = 21
Practice
In the following right triangles determine the value of x:
1.
2.
10
(x + 4)
12
x
2x
(x - 2)
x = 7.3
x = 5.4
Practice
The sides of a triangle are listed below, determine
whether the triangle is obtuse, acute or right.
1. 8, 9, 13
obtuse
2. 7, 12, 13
acute