Transcript Chapter 7: Proportions and Similarity

Chapter 7:
Proportions and Similarity
7.1- Proportions
 Make a Frayer foldable
7.1 Ratio
and
Proportion
Ratio
 A comparison of two quantities using division
 3 ways to write a ratio:



a to b
a
b
a:b
Proportion
 An equation stating that
two ratios are equal

Example: a  c
b d
 Cross products: means
and extremes

Example:
a c

b d
ad = bc
a and d = extremes
b and c = means
There are 480 sophomores and 520
juniors in a high school. Find the ratio of
juniors to sophomores.
Your Turn: solve these examples
Ex:
3 21

x 6
Ex:
x2 4

2
5
Your Turn: solve this example
 The ratios of the measures of three angles of
a triangle are 5:7:8. Find the angle measures.
A strip of wood molding that is 33 inches long is cut into
two pieces whose lengths are in the ratio of 7:4. What are
the lengths of the two pieces?
7.2 : Similar Polygons
 Similar polygons have:
 Congruent corresponding angles
 Proportional corresponding sides
A
Polygon ABCDE ~ Polygon LMNOP
B
L
E
M
C
D
P
N
Ex:
AB CD

LM NO
O
 Scale factor: the ratio of corresponding sides
If ΔABC ~ ΔRST, list all pairs of congruent angles and write a
proportion that relates the corresponding sides.
Determine whether the
triangles are similar.
A. The two polygons are similar. Find x and y.
If ABCDE ~ RSTUV, find the scale factor of ABCDE to RSTUV
and the perimeter of each polygon.
If LMNOP ~ VWXYZ, find the
perimeter of each polygon.
7.3: Similar Triangles
 Similar triangles have
congruent
corresponding angles
and proportional
corresponding sides
Z
Y
A
C
X
B
angle A  angle X
ABC ~
XYZ
angle B  angle Y
angle C  angle Z
AB AC BC


XY XZ YZ
7.3: Similar Triangles
 Triangles are similar if you show:

Any 2 pairs of corresponding sides are
proportional and the included angles are
congruent (SAS Similarity)
R
B
12
6
18
C
T
A
4
S
7.3: Similar Triangles
 Triangles are similar if you show:

All 3 pairs of corresponding sides are
proportional (SSS Similarity)
R
B
6
10
5
C
A
7
T
14
3
S
7.3: Similar Triangles
 Triangles are similar if you show:

Any 2 pairs of corresponding angles are
congruent (AA Similarity)
R
B
C
T
A
S
A. Determine whether the triangles are similar. If so, write a
similarity statement. Explain your reasoning.
B. Determine whether the triangles are similar. If so, write a
similarity statement. Explain your reasoning.
A. Determine whether the triangles
are similar. If so, write a similarity
statement. Explain your reasoning.
B. Determine whether the triangles are
similar. If so, write a similarity
statement. Explain your reasoning.
A. Determine whether the triangles are
similar. If so, choose the correct similarity
statement to match the given data.
B. Determine whether the triangles are
similar. If so, choose the correct
similarity statement to match the given
data.
ALGEBRA Given
, RS = 4, RQ = x + 3,
QT = 2x + 10, UT = 10, find RQ and QT.
SKYSCRAPERS Josh wanted to measure the height of the Sears
Tower in Chicago. He used a
12-foot light pole and measured its shadow at 1 p.m. The length
of the shadow was 2 feet. Then he measured the length of the
Sears Tower’s shadow and it
was 242 feet at the same time.
What is the height of the
Sears Tower?
7.4 : Parallel Lines and Proportional
Parts
 If a line is parallel to
one side of a triangle
and intersects the other
two sides of the
triangle, then it
separates those sides
into proportional parts.
A
Y
*If XY ll CB, then
AY AX

YC
XB
C
X
B
7.4 : Parallel Lines and Proportional
Parts
 Triangle Midsegment
Theorem

A midsegment of a
triangle is parallel to
one side of a triangle,
and its length is half of
the side that it is
parallel to
*If E and B are the midpoints
of AD and AC respectively,
1
then EB = 2 DC
A
E
D
B
C
7.4 : Parallel Lines and Proportional
Parts
 If 3 or more lines are
parallel and intersect
two transversals, then
they cut the
transversals into
proportional parts
A
B
C
D
E
F
AB DE

BC EF
AC BC

DF EF
AC DF

BC EF
7.4 : Parallel Lines and Proportional
Parts
 If 3 or more parallel
lines cut off congruent
segments on one
transversal, then they
cut off congruent
segments on every
transversal
If
AB  BC , then DE  EF
A
B
C
D
E
F
A. In the figure, DE and EF are midsegments of
ΔABC. Find AB.
B. Find FE.
C. Find mAFE.
MAPS In the figure, Larch, Maple, and Nuthatch Streets are all
parallel. The figure shows the distances in between city
blocks. Find x.
ALGEBRA Find x and y.
7.5 : Parts of Similar Triangles
 If two triangles are
X
similar, then the
perimeters are
proportional to the
measures of
corresponding sides
A
B
C
Y
perimeterABC AB BC AC



perimeterXYZ XY YZ
XZ
Z
7.5 : Parts of Similar Triangles
If two triangles are similar:
 the measures of the
 the measures of the
corresponding altitudes
are proportional to the
corresponding sides
X
corresponding angle
bisectors are
proportional to the
corresponding sides
A
S
M
B
C
D
Y
W
AD AC BA BC



XW XZ YX YZ
Z
L
R
O
N
U
MO MN LM LN



SU
ST
RS
RT
T
7.5 : Parts of Similar Triangles
 An angle bisector in a
 If 2 triangles are similar,
triangle cuts the opposite
side into segments that are
proportional to the other
E
sides
then the measures of
the corresponding
medians are
proportional to the
corresponding sides.
A
BC AB

CD AD
G
T
B
J
H
D
C
I
F
U
V
GI
GH
GJ
HJ



TV
UT
TW
UW
W
G
FG EF

GH EH
H
In the figure,
ΔLJK ~ ΔSQR. Find the
value of x.
In the figure, ΔABC ~ ΔFGH. Find the value of x.
Find x.
Find n.