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Triangles
9.2
The Pythagorean Theorem
The Pythagorean Theorem

In a right triangle, the sum of the legs
squared equals the hypotenuse squared.

a2 + b2 = c2, where a and b are legs and c
is the hypotenuse.
c
a
b
Pythagorean Triples

Pythagorean Triple
 When
the sides of a right triangle are all
integers it is called a Pythagorean triple.
 3,4,5
make up a Pythagorean triple since
3 2 + 4 2 = 5 2.
Example 1

Find the unknown side lengths. Determine
if the sides form a Pythagorean triple.
48
x
6
8
y
50
Example 2

Find the unknown side lengths. Determine
if the sides form a Pythagorean triple.
p
q
50
100
90
90
Example 3

Find the unknown side lengths. Determine
if the sides form a Pythagorean triple.
e
d
2
15
17
3
Example 4

Find the unknown side lengths. Determine if
the sides form a Pythagorean triple.
g
f
5 3
5
4 3
8
9.3
The Converse of the
Pythagorean Theorem
a
a
b
If a and b stay the same
length and we make the
angle between them
smaller, what happens to
c?
b
a
a
b
If a and b stay the same
length and we make the
angle between them
bigger, what happens to
c?
b
Classifying Triangles

Let c be the biggest side of a triangle, and
a and b be the other two side.
If c2 = a2 + b2, then the triangle is right.
 If c2 < a2 + b2, then the triangle is acute.
 If c2 > a2 + b2, then the triangle is obtuse.


*** If a + b is not greater than c, a triangle
cannot be formed.
Example 1
Determine what type of triangle, if any, can
be made from the given side lengths.
 7, 8, 12


11, 5, 9
Example 2
Determine what type of triangle, if any, can
be made from the given side lengths.
 5, 5, 5


1, 2, 3
Example 3
Determine what type of triangle, if any, can
be made from the given side lengths.
 16, 34, 30


9, 12, 15
Example 4
Determine what type of triangle, if any, can
be made from the given side lengths.
 13, 5, 7


13, 18, 22
Example 5

Determine what type of triangle, if any, can be
made from the given side lengths.
4, 8, 4 3

5, 5 2 , 5

9.4
Special Right Triangles
45º-45º-90º Triangles

Solve for each missing side. What
pattern, if any do you notice?
3
2
2
3
45º-45º-90º Triangles
5
4
4
5
45º-45º-90º Triangles
7
6
6
7
45º-45º-90º Triangles
½
300
300
½
45º-45º-90º Triangles
x
x
45º-45º-90º Triangles

In a 45º-45º-90º triangle, the hypotenuse is
times each leg.
x 2
x
x
2
30º-60º-90º Triangles

Solve for each missing length. What
pattern, if any do you notice?
10
10
10
30º-60º-90º Triangles
8
8
8
30º-60º-90º Triangles
6
6
6
30º-60º-90º Triangles
50
50
50
30º-60º-90º Triangles
2x
2x
2x
30º-60º-90º Triangles

In a 30º-60º-90º triangle, the hypotenuse is
twice as long as the shortest leg, and the longer
leg is 3 times as long as the shorter leg.
30º
2x
x 3
60º
x
Example 1

Find each missing side length.
6
45º
15
45º
Example 2
18
12
30º
45º
Example 3
30º
44
12
30º
30º
2x
x 3
60º
x
x 2
x
x
9.5
Trigonometric Ratios
Warm Up
Name the side opposite angle A.
 Name the side adjacent to angle A.
 Name the hypotenuse.

A
C
B
Trigonometric Ratios

The 3 basic trig functions and their
abbreviations are
 sine
= sin
 cosine = cos
 tangent = tan
SOH CAH TOA

sin =
opposite side
hypotenuse
SOH

cos = adjacent side
hypotenuse
CAH

tan =
TOA
opposite side
adjacent side
Example 1

Find each trigonometric ratio.
 sin A
A
 cos A
 tan A
5
3
 sin
B
 cos B
 tan B
C
4
B
Example 2

Find the sine, the cosine, and the tangent of the
acute angles of the triangle. Express each value
as a decimal rounded to four decimal places.
D
25
7
E
24
F
9.6
Solving Right Triangles
Example 1

Find the value of each variable. Round
decimals to the nearest tenth.
a
8
25º
Example 2
b
42º
40
Example 3
8
20º
c
Example 4
c
17º
10