triangles. 9.2 the pythagorean theorem in a right triangle, the sum of the legs squared equals the...
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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.
ac
b
Pythagorean Triples
Pythagorean TripleWhen the sides of a right triangle are all
integers it is called a Pythagorean triple.
3,4,5 make up a Pythagorean triple since 32 + 42 = 52.
Example 1
Find the unknown side lengths. Determine if the sides form a Pythagorean triple.
6
8
48
50yx
Example 2
Find the unknown side lengths. Determine if the sides form a Pythagorean triple.
100
q 90
90
50p
Example 3
Find the unknown side lengths. Determine if the sides form a Pythagorean triple.
2
3
e
1715
d
Example 4
Find the unknown side lengths. Determine if the sides form a Pythagorean triple.
4 3
5
g
8f
5 3
9.3
The Converse of the Pythagorean Theorem
a
c 2 = a 2 + b 2
b
a
b
If a and b stay the same length and we make the angle between them smaller, what happens to c?
If a and b stay the same length and we make the angle between them bigger, what happens to c?
a
c 2 = a 2 + b 2
b
a
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,
5, , 5
4 3
5 2
9.4
Special Right Triangles
45º-45º-90º Triangles
Solve for each missing side. What pattern, if any do you notice?
2
2
3
3
45º-45º-90º Triangles
4
4
5
5
45º-45º-90º Triangles
6
6
7
7
45º-45º-90º Triangles
300
300
½
½
45º-45º-90º Triangles
x
x
45º-45º-90º Triangles
In a 45º-45º-90º triangle, the hypotenuse is times each leg.
2
x
x2x
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 times as long as the shorter leg. 3
2x
x
30º
60º
3x
Example 1
Find each missing side length.
45º
15 45º
6
Example 2
45º
12
30º
18
Example 3
30º
12
30º
44
2x
x
30º
60º
3x
x
x2x
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 aresine = sincosine = cos tangent = tan
SOH CAH TOA
sin = opposite side hypotenuse
cos = adjacent sidehypotenuse
tan = opposite sideadjacent side
SOH
CAH
TOA
Example 1
Find each trigonometric ratio.sin Acos A tan Asin Bcos B tan B
3
4C
A
5
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.
7
24E
D
25
F
9.6
Solving Right Triangles
Example 1
Find the value of each variable. Round decimals to the nearest tenth.
25º
8a
Example 2
42º
40
b
Example 3
20º
c
8
Example 4
17º
10
c
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