9.1 – Similar Right Triangles
Theorem 9.1: If the altitude is drawn to the
hypotenuse of a right triangle, then the two triangles
formed are similar to the original triangle and to
each other.
Given : ABC with rt  ACB; altitude CN
Then : ACB ~ ANC ~ CNB
C
A
N
B
Theorem 9.2 (Geo mean altitude): When the altitude
is drawn to the hypotenuse of a right triangle, the
length of the altitude is the geometric mean between
the segments of the hypotenuse.
Given : ABC with rt  ACB; altitude CN
C
AN = CN
CN BN
A
N
B
Theorem 9.3 (Geo mean legs): When the altitude is
drawn to the hypotenuse of a right triangle, each leg
is the geometric mean between the hypotenuse and
the segment of the hypotenuse that is adjacent to
that leg.
Given : ABC with rt  ACB; altitude CN
C
AB = AC
AC AN
A
N
B
One way to
Theorem
9.3help
(Geo
remember
mean legs):
is thinking
When the
of it
altitude
as a car
is
drawn
and
you
todraw
the hypotenuse
the wheels.of a right triangle, each leg
is the geometric mean between the hypotenuse and
Another way is hypotenuse to hypotenuse, leg to leg
the segment of the hypotenuse that is adjacent to
that leg.
Given : ABC with rt  ACB; altitude CN
C
AB = BC
AC
AC AN
BC
BN
A
N
B
Set up Proportions
C
A
N
B
6 x

x 3
x 2  18
9 y

y 6
C
y  54
2
y 3 6
y
x
Geo legs
A
x3 2
Geo alt
N
6
z
3
w
6+3=9
w=9
9 z

z 3
z 2  27
B
z 3 3
Geo legs
w 15

15 9
9w  225
x  9  25
x  16
w  25
C
Geo legs
w
9
K
16 y

y 9
y 2  144
y  12
Geo alt
x
A
y
15
B
z
25 z

z 16
z 2  400
z  20
Geo legs
9.2 – Pythagorean Theorem
The Pythagorean Theorem: In a right triangle, the
square of the hypotenuse is equal to the sum of the
squares of the legs.
Given : ABC with rt  ACB
Then : a 2  b 2  c 2
a
b
c
Given
Starfish both sides
Cross Multiplication
(property of proportion)
Addition
Distributive Property =
Seg + post
Substituition prop =
• Pythagorean Triple is a set of three
positive integers a, b, and c that satisfy the
equation a2 + b2 = c2.
• Examples:
– 3, 4, 5
– 5, 12, 13
– 7, 24, 25
– 8, 15, 17
– Multiples of those.
6
y
13
12
x
12
x
5
8
12  8  x
2
2
2
144  64  x
2
208  x
2
4 13  x
5  x  13 9  12  y
2
25  x  169
15  y
2
14
9
2
2
2
2
2
x  144
x  12
2
DON’T BE FOOLED, no right angle
at top, can’t use theorems from
before
Find Area
8 in
9.3 – The Converse of the
Pythagorean Theorem
Converse of Pythagorean Theorem: If the square of
the hypotenuse is equal to the sum of the squares of
the legs, then the triangle is a right triangle.
Given : ABC with a  b  c
Then : ABC is a rt triangl e
2
2
2
C
a
B
b
c
A
Converse of Pythagorean Theorem: If the square of
the hypotenuse is equal to the sum of the squares of
the2legs,2 then
2 the triangle is a right triangle.
If c  a  b
Then mC  90; ABC
is2 acute
2
2
Given : ABC with a  b  c
2
2
2
If c : aABC
 b is a rt triangl e
Then
Then mC  90; ABC is obtuse
C
a
B
b
c
A
Is it acute, right, or obtuse (or neither)?
4, 11, 8
6, 8, 9
16 121 64
36 64 81
3 , 1, 2
3 1 4
5, 6, 12
5 + 6 < 12
Neither
+
<
Obtuse
+
>
Acute
+
=
Right
Watch out, if the sides are not in order, or are on a
picture, c is ALWAYS the longest side and should be
by itself
Is it acute, right, or obtuse (or neither)?
5, 6, 9
1, 2, 3
7 , 18 , 5
7, 7, 7
Reminders of the past. Properties of:
Parallelograms
Rectangles
1)
1)
2)
2)
3)
Rhombus
4)
1)
5)
2)
6)
3)
Describe the shape, Why? Use complete sentences
25
24
7
9.4 – Special Right Triangles
• Rationalize practice
30  60  90 Theorem
45   45   90  Theorem
In a 45   45   90  triangle,
is
the hypotenuse
In a 30  60  90 triangle, the hypotenuse
is 2 times as long as the short leg, and the
2 times as long as a leg
longer leg is 3 times the short leg
60 
45 
x
x 2
x
2x
30 
45 
x 3
x
Remember, small side with
small angle.
Common Sense: Small to big, you multiply (make bigger)
Big to small, you divide (make smaller)
For 30 – 60 – 90, find the smallest side first (Draw arrow to locate)
Lots of examples
Find areas
9.5 – Trigonometric Ratios
sine  sin
cosine  cos
Tangent  tan
Opposite
These
are
trig
ratios that
sin A 
Hypotenuse
describe the
ratio
adjacent
between
the
side
lengths
cos A 
Hypotenuse
given an angle.
tan A 
A device that helps is:
Opposite
adjacent
OPPOSITE
SOHCAHTOA
in pp yp os dj yp an pp dj
A
B
ADJACENT
C
B
8
2 15
A
2
C
sin A 
cos A 
tan A 
sin B 
cos B 
tan B 
• Calculator CHECK
– MODE!!!!!!!!!!! Should be in degrees
– sin(30o) Test, should give you .5
Find x
opposite, hypotenuse
Hypotenuse
20
USE SIN!
sin 34  
34
opposite
x
x
y
hypotenuse
20
. 5592 
x
20
11 . 184  x
Opposite
Pg 845
Angle
34o
sin
.5592
cos
tan
.8290 .6745
Or use the calculator
Look at what they want and
what they give you, then
use the correct trig ratio.
Find y
adjacent, hypotenuse
Hypotenuse
20
USE COS!
cos 34  
34
adjacent
y
x
y
hypotenuse
20
. 8290 
y
20
16 . 58  y
Adjacent
Pg 845
Angle
34o
sin
.5592
cos
tan
.8290 .6745
Or use the calculator
Look at what they want and
what they give you, then
use the correct trig ratio.
Find x
Adjacent, Opposite, use
TANGENT!
tan x 
30
opposite
30
x
adjacent
4
4
tan x  7 . 5
Adjacent
Opposite
x  82 
Pg 845
Angle sin
81o
82o
83o
cos
tan
.9877 .1564 6.3138
.9903 .1392 7.1154
.9925 .1219 8.1443
If you use the calculator,
you
putthey
tan-1want
(7.5) and
and
Lookwould
at what
it
will they
give give
you an
angle
what
you,
then
back.
use the correct trig ratio.
20
x
50
8
x
83
12
6
49
x
41
x
8
x
y
6
20
50
y
x
40
x
70
70
17
34
cos 70  
17
x
For word problems, drawing a picture helps.
From the line of
sight, if you look up,
it’s called the ANGLE
OF ELEVATION
ANGLE OF ELEVATION
ANGLE OF DEPRESSION
From the line of sight, if
you look down, it’s
called the ANGLE OF
DEPRESSION
All problems pretty much involve trig in some way.
Mr. Kim’s eyes are about 5 feet two inches above the
ground. The angle of elevation from his line of sight
to the top of the building was 25o, and he was 20 feet
away from the building. How tall is the building in
feet?
25 
x
20 feet
tan 25  
x
20
x  9 . 326
 5.167  14.493
5.167
Mr. Kim is trying to sneak into a building. The
searchlight is 15 feet off the ground with the beam
nearest to the wall having an angle of depression of 80o.
Mr. Kim has to crawl along the wall, but he is 2 feet wide.
Can he make it through undetected?
80o
tan( 80 ) 
15
x
 2 . 644 ft
Mr. Kim saw Mr. Knox across the stream.
He then walked north 1200 feet and saw
Mr. Knox again, with his line of sight and
his path creating a 40 degree angle. How
wide is the river to the nearest foot?
1200 ft
x
tan( 40 ) 
1200
 1007
ft
The ideal angle of elevation for a roof for effectiveness
and economy is 22 degrees. If the width of the house is
40 feet, and the roof forms an isosceles triangle on top,
how tall should the roof be?
• DJ is at the top of a right triangular block of
stone. The face of the stone is 50 paces long.
The angle of depression from the top of the
stone to the ground is 40 degrees (assume DJ’s
eyes are at his feet). How tall is the triangular
block?
9.6 – Solving Right Triangles
Find x
Adjacent, Opposite, use
TANGENT!
tan x 
30
opposite
30
x
adjacent
4
4
tan x  7 . 5
Adjacent
Opposite
x  82 
Pg 845
Angle sin
81o
82o
83o
cos
tan
.9877 .1564 6.3138
.9903 .1392 7.1154
.9925 .1219 8.1443
If you use the calculator,
you
putthey
tan-1want
(7.5) and
and
Lookwould
at what
it
will they
give give
you an
angle
what
you,
then
back.
use the correct trig ratio.
Find x
Find all angles and sides, I check HW
Find all angles and sides