Download Chapter 8 Notes - cloudfront.net

Chapter 8 Notes
Mrs. Myers – Geometry
Name ______________________________
Period ______
8.1 Ratio & Proportion

a
or a : b
b
o must be the _____________ units
o always _____________________
o b0
Ratio of a to b:
Conversion of unit measurements







100 cm = 1 m
12 in = 1 ft
1000 m = 1 km
3 ft = 1 yd
1000 g = 1 kg
5280 ft = 1 mi
16 oz = 1 lb
Ex. 1 Simplify the ratios
A)
12 cm
4m
B)
6 ft
18 in
C)
1m
2 km
D)
3 yd
6 ft

Using Ratios:
o Working backwards –
p  2l  2 w
p  60 cm
ab : bc 
ab
bc
Ex. 2 The perimeter of the isosceles triangle shown is 56 in. The ratio of lm : mn is
5: 4 . Find the lengths of the sides and the base of the triangle.

Using extended Ratios:
o Working backwards –
m 1  m 2  m 3  180
30 : 60 : 90
Ex. 3 The measures of the angles in a triangle are in the extended ratio 3: 4:8 . Find the
measures of the angles.
Ex. 4 The ratio of the side lengths of DEF to the corresponding side lengths of
ABC are 2 :1. Find the unknown lengths.

Proportion: an equation that equates _________ ratios.
a
c
a c
o If the ratio
is equal to the ratio , then 
b
b d
d



Extremes = numbers ____ and ____
Means = numbers ____ and ____.
Properties of Proportions:
o Cross Product Property = the product of the extremes ________ the
product of the means.
a c
 , then ad  bc
 If
b d
o Reciprocal Property = if 2 ratios are equal, then their reciprocals are equal.
a c
b d
 , then

 If
b d
a c
Ex. 5 Solve the proportions
A)
9
6

14 x
B)
2
4

x3 x
C)
x 5
x

4
10
Ex. 6 The ratios are given. Solve for the variable.
A) The ratio bc : dc is 2 : 9 . Find x.
B) The ratio ba : ca is 3 : 9 . Find x.
B
A
B
2
x
D
27
C
A
x
C
8.2 Problem Solving in Geometry with Proportions

Additional Properties of Proportions:
a c
a b
If
 , then

b d
c d
o
d c
then

b a
If
o
a c
ab cd
 , then

b d
b
d
a b c d
then

b
d
Ex. 1 using properties of proportions, state whether the statement is true or false.
A) If
p
r
p 3
 , then

6 10
r 5
B) If
x 15
x 3
 , then

10
y
y 2
C) If
a c
a 3 c3
 , then

3 4
3
4
D) If
3 5
3 x 5 y
 , then

x y
x
y
Ex. 2

ab ac

. Find bd .
bd ce
Geometric Mean: where a, b  0 & x  0 such that
a x

 ___________
x b
Ex. 3 Find the geometric mean.
A) 3 and 27
B) 8 and 20
8.3 Similar Polygons

Similar Polygons: corresponding angles are ___________________ and lengths
of corresponding sides are _____________________________.
ab bc cd da
o ie : ABCD EFGH where



ef
fg gh he
Ex. 1 Pentagons JKLMN and STUVW are similar. List all congruent angles. Then
write the ratios of the corresponding sides.
Ex. 2 Are the figures similar? If they are similar, write a similarity statement.
A)
B)
Ex. 3 You have been asked to create a poster. You have a 3 in by 4in photo (l x w).
You need to enlarge it to be 20 in. wide. How long will it be?

Scale Factor: the ratio of the lengths of 2 corresponding sides when 2 polygons
are similar (doesn’t matter which corresponding sides, since they will be equal).
Ex. 4 Calculate the scale factor of the pool to the patio. Then find the ratio of their
perimeters  pool patio  .
* Theorem 8.1: If 2 polygons are similar, then the ratio of their perimeters is _________
to the ratios of their corresponding side lengths.
If KLMN PQRS , then
kl  lm  mn  nk
kl
lm mn nk




pq  qr  rs  sp
pq qr
rs
sp
Ex. 5 JKLM PQRS . Find x.
8.4 Similar Triangles
Ex. 1
LM
LMN
PQN Given
QP, m Q  36, m LNM 106, QN  20, LM  33, and QP  30.
L
M
N
Q
P
A) Write the statement of proportionality.
B) Find m M and m P .
C) Find MN and QM.
* Postulate 25: Angle-Angle (AA) Similarity Postulate = if 2 angles of a triangle are
___________ to 2 corresponding angles of another triangle, then the triangles are similar.
If K  Y and
J  X , then ___________________
Ex. 2 Explain why WVX

WZY
Note: If 2 polygons are similar, then the ratio of any two corresponding lengths
(altitudes, medians, angle bisectors segments, and diagonals) is equal to the scale
factor of the similar polygons.
Ex. 3 Find the length of EH given: CF  6, AD  5, DB  7.5 and
E
C
A
F
D
H
B
ACD  AEB .
8.5 Proving Triangles are Similar
* Theorem 8.2: Side-Side-Side (SSS) Similarity Theorem =
If
ab bc ca


, then ____________________
pq qr rp
* Theorem 8.2: Side-Angle-Side (SAS) Similarity Theorem =
If
X  M and
zx
xy

, then
pm mn
Ex. 1 Which of the following 3 triangles are similar?
XYZ
MNP
Ex. 2 A pantograph is used to draw an enlargement of a daisy. In the diagram,
pb
pa
.

pd
pc
A) Why is PBA
PDC ?
B) In the diagram, PA = 8 in and AC = 8 in. the Diameter of the original daisy is 1.8 in.
What is the diameter of the daisy in the enlargement?
Ex. 3 At an indoor climbing wall, a person whose eyes are 6 ft from the floor places a
mirror on the floor 60 ft from the base of the wall. Then he walked backwards 5 ft before
seeing the top of the wall in the center of the mirror. Use similar triangles to estimate the
height of the wall.
x ft
Center of
mirror
6 ft
5 ft
60 f t
Ex. 4 Use the given lengths to find the width LM of the river.
N
8 ft
L
x ft
M
5 ft
42 f t
P
Q
8.6 Proportions and Similar Triangles
* Theorem 8.4: Triangle Proportionality Theorem = If a line parallel to one side of a
triangle intersects the other 2 sides, then it divides the 2 sides proportionally.
If TU
QS , then
rt ru
rt ru

or

tq us
rq rs
* Theorem 8.5: Converse of the Triangle Proportionality Theorem = If a line divides 2
sides of a triangle proportionally, then it is parallel to the third side.
If
rt ru
rt ru

or

, then TU
tq us
rq rs
QS
Ex. 1 In the diagram UY VX , UV  3, UW 18, and XW  25. What is the length of
YX ?
U
V
W
X
Y
Ex. 2 Given the diagram, determine whether PQ TR ?
* Theorem 8.6: If R S and S T and L and M intersect R, S and T, then
R
S
uw vx
.

wy xz
T
u
w
y
v
x
z
L
M
* Theorem 8.7: If CD bi sec ts
ACB , then
ad ca

.
db cb
A
D
B
C
Ex. 3
1
2
3 . What is x?
6
9
1
8
x
Ex. 4
3
2
LKM 
MKN , KL  3, KN 17 , LN 15. Find the length of MN .
L
M
N
K
Ex. 5 FJ GI , FG  2, GH  8, FJ  9, and HJ 12. . Find x and y.
F
G
H
y
x
J
I
8.7 Dilation

A dilation with center C and scale factor k is a transformation that maps every
point P in the plane to a point P’ so that the following properties are true:
o If P is not the center point C, then the image point P’ lies on CP . The
CP '
scale factor k is a positive number such that k 
, and k  1 .
CP
o If P is the center C, then P = P’.

A dilation is______________________________ if 0  k  1

A dilation is ______________________________ if k  1
Ex. 1 Find the scale factor (k) and identify the dilation (reduction or enlargement).
A)
B)
C)
D)
E)
NOTE: When C is the origin, coordinates of each point:
image  preimage scale factor
F)
Ex. 2 When the origin is the center of the dilation, what are the coordinates of the image
of
ABC given the specified scale factor?
A) k  1.5
B) k  3
C) k 
1
3
Ex. 3 Identify the dilation and find the scale factor. Then find the value of the
variable(s).
A)
B)