Download Right Triangles and Trigonometry

Right Triangles and
Trigonometry
Chapter 8
8.1 Geometric Mean
 Geometric mean:

Ex: Find the
geometric mean
between 5 and 45
# x

x #

Ex: Find the
geometric mean
between 8 and 10
5
x

x
45
5  45  x  x
8
x

x 10
8 10  x  x
225  x 2
80  x 2
225 
15  x
x2
80 
x2
4 5  8.9  x
 If an altitude is drawn from the right angle of a
right triangle. The two new triangles and the
original triangle are all similar.
B
A
D
C
ABC ~ ADB ~ BDC
 The altitude from a
right angle of a right
triangle is the
geometric mean of
the two hypotenuse
segments
Ex:
AD BD

BD DC
hyp.seg .1 altitude

altitude hyp.seg .2
B
A
D
C
 The leg of the
hypotenuse
leg
triangle is the

leg
adjacent.hyp.seg .
geometric mean of
the hypotenuse and
the segment of the
B
hypotenuse adjacent
Ex:
AC AB

AB AD
AC BC

BC DC
A
D
C
8.2 Pythagorean Theorem and
its Converse
 How, when and why do you use the
Pythagorean Theorem and its converse?



How: the square of the two legs added
together equals the hypotenuse squared
When: given a right triangle and the length
of any two sides
Why: to find the length of one side of a
right triangle
Pythagorean Theorem:
a b  c
c
a
2
2
2
b
 When c is unknown:
 When a or b is unknown:
5 2  32  x 2
x
5
3
25  9  x 2
7 2  x 2  14 2
14
7
34  x 2
34  x
x  5.8
49  x 2  196
x 2  147
2
x
x 2  147
x  12.1
 Converse: the sum of
the squares of 2
sides of a triangle
equal the square of
the longest side
8, 15, 16

8  15  16
64  225  256
2
2
2
289  256
Not =, so not a right triangle
 Pythagorean Triple:

3 lengths that always
make a right triangle
3, 4, 5
 5, 12, 13
 7, 24, 25
 9, 40, 41

8.3 Special Right Triangles
 30-60-90


Short leg is across from
the 30 degree angle
Long leg is across from
the 60 degree angle
Ex:
x  2  14
x7
14
x
30
y
y7 3
shortleg  3  longleg
shortleg  2  hypotenuse
 45-45-90

Ex:
x
The legs are
congruent
x 2  6
6
x
leg  2  hypotenuse
Ex:
6
2
x
6
2
x

2 2
x
x
6 2
3 2
2
8
x 8 2
8.6 Law of Sines
B
c
SinA SinB SinC


a
b
c
a
A
C
b
 Use two of the ratios to make a proportion and
solve
 To solve a triangle: means to solve for all
missing angles and sides
 Solve the triangle
A
33
14
B
47
C
8.7 Law of Cosines
B
c
a
A
C
b
a 2  b 2  c 2  2bc cos A
b 2  a 2  c 2  2ac cos B
c 2  a 2  b 2  2ab cos C
 Solve the Triangle
A
60
8
10
C
B