Download Ch 6

Mth 97
Fall 2013
Chapter 6
Section 6.1 – Ratio and Proportions
Ratios
_____________________ geometric figures have the same shape, but not necessarily the same size.
A _________________ is an ordered pair of numbers. The ratio of the number a to b may be represented in
three ways:
For the rectangle to the right, calculate the following ratios. Give
a.
length to width
15 in
36 in
Since in our ratios the units were the same, they cancelled. If the units are not the same, then our
ratio would be called a ______________. In application problems we sometimes need to use
dimensional analysis to convert to equivalent rates.
b. width to length
c. length of the diagonal to the perimeter
d. perimeter to area
Proportions
A proportion is and equation stating that two ratios are _______________.
a c
 is a proportion in which a, b, c, and d are called _____________.
b d
a and d are called the ____________________.
b and c are called the ____________________.
1
Mth 97
Fall 2013
Chapter 6
Show that each equation is a proportion by using equal fractions.
1 3

2 6
1.2 2

1.8 3
π: 5π = 1:5
9x 9

4x 4
Theorem 6.1 – Cross Multiplication Theorem
In a proportion the product of the means equals the product of the extremes.
a c
 , if and only if, ad = bc.
b d
Use cross products to show that the following are or ore not proportions.
15 21

35 49
15 35

21 49
35 49

15 21
49 35

15 21
Theorem 6.2 – Summarizes the above findings
a c

if and only if
b d
a c

if and only if
b d
a c

if and only if
b d
a b
 .
c d
d c

b a
b d

a c
(Exchange the means)
(Exchange the extremes)
(Invert each ratio)
You can use cross products to solve for an unknown in a proportion.
16 48

17 x
15 42

8
x
x  2 16

15
40
x5 x

11
6
2
Mth 97
Fall 2013
Chapter 6
A rectangular snapshot measures 3.5 inches tall by 5 inches wide. An enlargement of the photo will be made,
where the larger photo will have a width of 18 inches. If none of the original picture will be cropped, the
dimensions of the enlargement are proportional to the original photo. What will be the height of the larger
photograph?
Theorem 5.3 – If
a b
 , then b is the geometric mean (or mean proportional) of a and c.
b c
Find the geometric mean of 4 and 9.
Find the mean proportional of 5 and 10.
Do ICA 10, problems 1 and 2
Section 6.2 – Similar Triangles
Triangles are similar if all corresponding angles are _____________________ and all corresponding
sides are _________________________________.
A
6 cm
12cm
R
8 cm
4 cm
C
15 cm
∆ABC
B
T
10 cm
S
∆RST means: A  R , B  S , and C  T
and
AB BC CA


.
RS ST TR
The following properties that hold true for congruent triangles, also hold true for similar triangles.
1.
Reflexive Property
2. Symmetric Property
3. Transitive Property
3
Mth 97
Fall 2013
Triangle Similarity:
Chapter 6
Once you establish that two triangles are similar, you know all three pairs of
corresponding angles are congruent and all three pairs of corresponding sides are proportional.
AAA Similarity
E
B
You know three angles of one triangle are
congruent to three angles in another triangle.
C D
A
F
If __________________________________
AA Similarity
E
B
Conclude: ABC  DEF
You know two angles in one triangle are congruent
to two angles in another triangle.
A
C
D
F
If ___________________________________
B
Conclude: ABC  DEF
SAS Similarity
E
c
A
You know one pair of angles is congruent and two
sides of one triangle are proportional to two sides
of the other triangle.
f
C
b
D
e
F
If _____________________________________
B
c
A
b
SSS Similarity
E
a
f
d
C
D
e
Conclude: ABC  DEF
F
If _____________________________________
You know nothing about the angles. You do know
that three sides of one triangle are proportional to
the three sides of another triangle.
Conclude: ABC  DEF
4
Mth 97
Fall 2013
Chapter 6
Name two similar triangles and state which similarity property applies. Find the missing lengths.
H
T
12
x
9
A
y
y
15
6
HM TR
M
A
R
x
B
5
6
3
E
y
D
z
C
Finish ICA 10.
A
Section 6.3 – Applications of Similarity
Name three pairs of similar triangles.
D
A
30°
0
60°
C
B
Results for any Right Triangle
D
(1) The height BD to the hypotenuse is the
geometric mean of AD and DC.
C
B
ABC is a right triangle. CD is the altitude of the
triangle from vertex C (also called the height).
AD CD

orCD  ( AD)( DB)
.
CD DB
(2) The three triangles formed are all similar to
each other. ABC  ACD  CBD
5
Mth 97
Fall 2013
Chapter 6
R
Using the Mean Proportional
Name the three similar triangles.
If UT= 5 and US = 8, find AC.
T
U
S
Side Splitting Theorem
B
DE AC
You have overlapping triangles so that the bases
are parallel, and the triangles share an angle. By
AAA Similarity, we know ABC  DBE. Another
D
E
A
useful result is that DE cuts the two sides of the
large triangle into proportional segments.
C
Conclude:
If _________________________________
AD CE

DB EB
Using the Side Splitting Theorem
B
C
If AB = 5, BC = 2, and AE = 7, find ED.
A
If AC = 10, AE = 8, and ED = 4, find BC.
E
D
D
If BE = 7, DC = 10, and AE = 2 + ED, find AE.
J
I
E
F
If DJ = 5, JI = 3, DE = 8, and FG = 7, find IH and EF.
H
G
6
Mth 97
Fall 2013
Chapter 6
Midsegment Theorem
B
In the special case where D is the midpoint of AB ,
DE AC
D
and E is the midpoint of CB , the lengths of the
bases of the two triangles have a special
relationship. DE is half of AC.
E
C
A
If _____________________________________
Conclude: DE  12 AC
Using the Midsegment Theorem
A
If DC = 16, find EB.
E
B
D
C
Corollary 6.12 – Midquad Theorem
If ABCD is any quadrilateral and E, F, G, H are midpoints as shown then EFGH is a parallelogram.
A
F
E
B
D
H
G
C
Do ICA 11, problems 1 and 2
7
Mth 97
Fall 2013
Chapter 6
Section 6.4 – Using Right Triangle Trigonometry to Solve Geometry Problems
A
Trigonometry is the study of measures of ____________________.
Definitions of Trig Ratios (for the acute angles in a right triangle)
b
sine(acute angle) = length of opposite leg
length of hypotenuse
Sin A =
Sin B =
cosine(acute angle) = length of adjacent leg
length of hypotenuse
Cos A =
Cos B =
tangent(acute angle) = length of opposite leg
length of adjacent leg
Tan A =
Tan B =
A way to remember these is:
SOH
CAH
c
C
a
B
TOA
Compute the following. Give answers rounded to 2 decimal places.
cos A, if b = 7 and c = 12
sin B, if a = 5 and c = 12
tan B, if a = 5 and c = 13
Using a calculator to find the sine, cosine, and tangent ratios of angles
To find tan 43° (check that your calculator is set in degrees, not radians)
Press Tan 43 ENTER. Your result should be 0.9325150861 which rounds to 0.9325 (4 decimal places).
sin 43° =
cos 43° =
Compute the following. Give answers rounded to 2 decimal places.
Find b, if mA  27.3 and c = 5.
Find c, if mB  42 and b = 10.
8
Mth 97
Fall 2013
Chapter 6
At a horizontal distance of 150 feet from the base of a building, the line of sight to the top of the building
makes an angle of 21°with level ground. That means the angle of elevation to the top of the building is 21°.
How tall is the building to the nearest hundredth of a foot? Assume the building is perpendicular to the
ground.
Use Inverse trigonometric functions to find the approximate measure of an acute angle in a right triangle
If you know a trig ratio in a right triangle, you can find the measures of both acute angles by using sin-1, cos-1,
or tan-1. We usually round to tenths of a degree.
If tan A = 0.8391, enter tan-1 (2nd tan) .8391
_________________
mA 
If cos A = 0.9063, enter cos-1 (2nd cos) .9063
_________________
mA 
If sin A = 0.3581, enter sin-1 (2nd sin) .3581
_________________
mA 
Given a right triangle with right angle C with sides
4, and 5, find the measure of angles A and B.
measuring 3,
B
A
C
Theorem 6.13 – In a right triangle with acute angle A, sin A divided by cos A is tan A.
sin A
 tan A
cos A
Verify this theorem using the right triangle below.
5A
sin A =
cos A =
tan A =
13
5
C
12
B
sin A

cos A
9
Mth 97
Fall 2013
Chapter 6
Theorem 6.14 – In a right triangle with acute angle A, the sum of sin2A and cos2A is 1.
sin 2 A  cos 2 A  1
This theorem is verified for a right triangle on page 335.
Given that sin A = 0.6428, find the cos A.
Solving Right Triangles
To solve a right triangle means to find all the missing sides and angles.
B
12”
To find mA
To find mB
To find b use tan
or use Pythagorean Theorem
4”
C
b
We know mC  90
A
a=4
c = 12
Solve each right triangle.
E
20.43 m
8 cm
C
8.72 m
F
G
Given mG  45
mF  90
EF = 8
A
B
Given a = 8.72
b = 20.43
mC  90
Find mE 
Find c =
FG =
mA 
EG =
mB 
10
Mth 97
Fall 2013
B
Chapter 6
∆ABC is not a right triangle. Determine the area of ∆ABC given that
mA  78 , AB = 8 mm and AC = 17 mm. Hint: First draw a
perpendicular segment from angle B to segment AC to find the
triangle’s height (altitude).
A
C
Finish ICA 11
Section 6.5 – Using Laws of Trigonometry to Solve Geometry Problems
An _______________________ triangle is a triangle with no right angles. Oblique triangles can be acute or
obtuse, but we will only discuss acute triangles.
Theorem 6.15 – The Law of Sines
sin A sin B sin C


a
b
c
*Use this to find the length of a side or the measure of an acute
A
angle in an acute triangle.
or
a
b
c


sin A sin B sin C
c
b
C
a
B
To use this law you must know the measure of at least one side and the measures of any two acute angles or
the measures of any two sides and the measure of the angle opposite one of them.
Use the Law of Sines to help solve the triangles with the given information below.
Given: mA  27 , mC  70 , and AB = 10
B
Find: mB , AC, and BC
A
C
Given : mF  47.6 , mG  61 , and g  10 2
E
Find: mE , f, and e
G
F
11
Mth 97
Fall 2013
H
Chapter 6
Given: mH  78 , HI = 8 cm, and JI = 22 cm.
Find: mI , mJ , and JH
J
I
Application Example (See diagram on page 349, number 44)
While whale watching off the Oregon coast, two observers standing 65 feet apart on the deck of a boat
estimate the angle from boat to whale to be 35° and 85° respectively. How far is the whale from each
observer?
Theorem 6.16 – The Law of Cosines
a 2  b 2  c 2  2bc cos A
B
c
A
b 2  a 2  c 2  2ac cos B
a
b
C
c 2  a 2  b 2  2ab cos C
Use this when you know the measure of two sides and the included angle or the measure of all three sides.
Use the Law of Cosines to solve the following problems.
If mA = 85°, b = 36.2 and c =54.6, find the side length a in the triangle.
B
A
C
12
Mth 97
Fall 2013
Chapter 6
If a = 22, b = 21 and c = 24 find mB . ( b 2  a 2  c 2  2ac cos B )
B
A
C
Two airplanes take off from the same airport. At a certain time, one plane is 200 miles from the airport and
the other is 250 miles from the airport. If the angle between them is 70°, how far apart are the planes? Give
your answer to the nearest whole number of miles. (Hint: first draw a diagram.)
13