Download 7.0 Rationalizing the Denominator You do not leave a radical in the

7.0 Rationalizing the Denominator
You do not leave a radical in the denominator of a fraction.
You must rewrite it as an equivalent fraction with a rational number in
the denominator. Hence the name – rationalizing the denominator.
1. Simplify the radicand first.
2. Multiply the numerator and the denominator by a factor (a form of 1)
that eliminates the radical in the denominator.
EExample 1: Simplify
Multiplying the top and bottom by
will create the
smallest perfect square under the square root in the
denominator.
Replacing
by 7 rationalizes the denominator.
It is ok to have an irrational number in the top (numerator) of a fraction.
Remember: Anything divided by itself is just 1, and multiplying by 1 does not
change the value of whatever you're multiplying by the 1
1)
1 =
√5
2)
√35 =
√15
7.1 Geometric Mean
The geometric mean of two positive numbers a and b is the positive number
x such that a:x = x:b is true.
If you solve this proportion for x, you find that x = √ab which is a positive
number.
Find the geometric mean of 8 and 18.
8= x
x 18
x2 = 144
x = 12
Short cut: x = √8∙18
x = √144
x = 12
For example, the geometric mean of 8 and 18 is 12, because
x = √8 ∙ 18
x = √144
x = 12
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.
C
A
B
D
∆CBD ~ ∆ABC
∆ACD ~ ∆ABC
S
 B
 BCD
CDB
 A
ACD
L

 A



S
____ ____ ____
ADC
ACB
_____ ____
____
L
 In right ∆ABC, altitude CD is drawn to the hypotenuse, forming two
smaller right triangles that are similar to ∆ABC, you know that
∆CBD ~ ∆ACD.
C
S
BCD
L

S
BD =
L
B

CD
CDB

BC
A
CD
AD
D
AC
B
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.
C
A
D
∆CBD ~ ∆ABC
S
 B
 BCD
L
 B
 A
S
CD
L
AC
B
∆ACD ~ ∆ABC
 A
ACD
ADC
 A
ABC
ACB
BD = BC
CD
AD
BC
BC
AC
CDB
 ACB
AB
=
AC
AB
 In right ∆ABC, altitude CD is drawn to the hypotenuse, forming two
smaller right triangles that are similar to ∆ABC, you know that
∆CBD ~ ∆ACD.
C
S
BCD
L
A
S
BD =
L
CD
B
ACD
CDB
ADC
CD
BC
AD
AC
A
CD
AD
D
AC
B
In a right triangle, the altitude drawn from the right angle to the
hypotenuse divides the hypotenuse into two segments.
The length of the altitude is the geometric mean between the
segments of the hypotenuse. Memorize this.
C
A
D
B
C
1) Find CD, if AD = 10 and DB = 4
A
2) Find AD if CD = 12 and DB = 3
D
B
In a right triangle, the altitude from the right angle to the hypotenuse
divides the hypotenuse into two segments.
 The length of each leg of the right triangle is the geometric mean
between the hypotenuse and the segment of the hypotenuse that is
adjacent to the leg. Memorize this.
C
A
D
B
C
1) Find AC, if AD = 6 and DB = 4
A
2) Find BC, if AD = 10 and DB = 6
D
B
7.2 Pythagorean Theorem
• In a right triangle, the square of the length of the hypotenuse
is equal to the sum of the squares of the legs.
c
a
b
c2 = a2 + b2
The Converse of the Pythagorean Theorem is also true
• If the square of the length of the longest side of the triangle is equal
to the sum of the squares of the lengths of the other two sides, then
the triangle is a right triangle.
•
• If c2 = a2 + b2, then ∆ABC is a right triangle.
B
c
a
C
b
A
A Pythagorean Triple
The most popular right triangle is the 3, 4, 5 triangle.
A triangle with sides 3, 4, 5 is a right triangle.
The second most popular right triangle is a 5, 12, 13 triangle.
Are you a right triangle?
3
4
5
6
8
10
9
12
15
12
16
20
15
20
25
Any multiple of a 3, 4, 5 triangle is also a right triangle.
Are you a right triangle?
5
12
13
10
24
26
15
36
39
20
48
52
Any multiple of a 5, 12, 13 triangle is also a right triangle.
Acute, Obtuse, or Right Triangle?
First check to see if the 3 sides given form a right triangle.
Remember: The sum of 2 sides must be greater than the 3rd side.
• If c2 = a2 + b2, then ∆ABC is a right triangle.
If sides are 6, 8, 10, is it a right triangle? If not, what is it?
• If c2 < a2 + b2, then ∆ABC is an acute triangle.
If sides are 7, 8, 10, is it a right triangle? If not, what is it?
• If c2 > a2 + b2, then ∆ABC is an obtuse triangle.
If sides are 4, 5, 8, is it a right triangle? If not, what is it?
1) 8, 5, 5
2) 4, 4, 5
3) 6, 10, 18 – Watch it
30˚- 60˚- 90 ˚ Right Triangle
c2 = a2 + b2
22 = 12 + b2
1
4 - 1 = b2
b
60˚
30˚
b = √3
2
c2 = a2 + b2
2
b
4
c2 = a2 + b2
3
b
6
c2 = a2 + b2
4
b
8
Isosceles Right Triangle
c2 = a2 + b2
c2 = 12 + 12
1
1
c2 = 3
c = √2
C
c2 = a2 + b2
2
2
C
c2 = a2 + b2
3
3
C
c2 = a2 + b2
4
4
C
7.3 Special Right Triangles
Isosceles Right Triangle
Leg
Leg
Hyp
Equilateral Triangle
SLeg
LLeg
Hyp
7.3B Special Right Triangles
There are special properties for right triangles whose angle measures are
45°- 45°- 90° or 30°- 60°- 90°.
The 45°- 45°- 90° triangle is the only type of isosceles right triangle.
In a 45°- 45°- 90° triangle, the hypotenuse is √2 times as
long as each leg. The ratio of the sides is 1:1:√2.
√2x
x
x
Hypotenuse = √2 ∙ leg
• In a 30°- 60°- 90° triangle, the hypotenuse is twice as long as the
shorter leg, and the longer leg is √3 times as long as the shorter leg.
2x
x
√3x
Hypotenuse = 2∙shorter leg
Longer leg = √3∙shorter leg
7.4 Trigonometry
• A trigonometric ratio is a ratio of the lengths of two sides of a right
triangle. The word trigonometry is derived from the ancient Greek
language and means measurement of triangles. The three basic
trigonometric ratios are sine, cosine, and tangent, which are
abbreviated as sin, cos, and tan respectively.
• Let ∆ABC be a right triangle. The sine, the cosine, and the tangent of
the acute angle A are defined as follows.
•
B
hypotenusec
A
b
side adjacent to angle A
Side
a opposite
angle A
C
SOH COA TOA
Sin A = side Opposite A
Hypotenuse
Cos A = side Adjacent to A
Hypotenuse
Tan A = side Opposite A
side Adjacent to A
The value of this ratio does not depend on the size of the triangle
or the measures of the sides.
Trig ratios are used to find missing measures of a right triangle.
You only need to know the measures of two sides or the measures
of one side and one acute angle.
The value of a trigonometric ratio depends only on the measure
of the acute angle, not on the particular right triangle that is
used to compute the value.
Large
Sin A opposite
hypotenuse
Small
8/17 =
Cos A adjacent
hypotenuse
Tan A opposite
adjacent
B
B
17
8
8.5
4
A
15
C
A
7.5
C
The value of a trigonometric ratio depends only on the measure
of the acute angle, not on the particular right triangle that is
used to compute the value.
Large
Small
Sin A opposite
hypotenuse
8/17 = .470588
4/8.5 = .470588
Cos A adjacent
hypotenuse
15/17 = .88235
7.5/8.5 = .88235
Tan A opposite
adjacent
8/15 = .533333
4/7.5 = .533333
Trig ratios are often expressed as decimal approximations.
You will notice that the sine or the cosine of an acute triangle is
always less than 1. The reason is that these trigonometric ratios
involve the ratio of a leg of a right triangle to the hypotenuse. The
length of a leg or a right triangle is always less than the length of
its hypotenuse, so the ratio of these lengths is always less than
one.
Ex. If the hypotenuse is 6, then a leg must be less than 6. Let’s
use 2. So the ratio is the leg/hyp or 2/6. The ratio is always less
than 1.

If you hit the sine, cosine, or tangent key, remember to change radians to degree.
You can use a calculator to approximate the sine, cosine, and the
tangent of 74.
Make sure that your calculator is in degree mode.
Sample keystroke
sequences
Sample calculator
display
Rounded
Approximation
Sample keystroke
sequences
Sample calculator
display
Rounded
Approximation
Sin (74)
0.961262695
0.9613
Cos (74)
0.275637355
0.2756
Tan (74)
3.487414444
3.4874
Sin (74)
Cos (74)
Tan (74)
7.5 Angles of Elevation and Depression
• Suppose you stand and look up at a point in the distance.
Maybe you are looking up at the top of a tree. The angle that
your line of sight makes with a line drawn horizontally is called
angle of elevation.
• You are measuring the height of a Sitka spruce tree in Alaska.
You stand 45 feet from the base of the tree. You measure
the angle of elevation from a point on the ground to the top of
the top of the tree to be 59°. To estimate the height of the
tree, you can write a trigonometric ratio that involves the
height h and the known length of 45 feet.
7.6
Use “Zorro” to find the angle of depression. It is the angle
formed by the horizontal line and the line of sight.