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

b

a

Ex: There are 480 sophomores and 520 juniors in a high school. Find the ratio of juniors to sophomores. Then find the ratio of sophomores to students.

Proportion

An equation stating that two ratios are equal Example:

Cross products: means and extremes Example:

d

c

b

a

d

c

b

a a and d = extremes

b and c = means

ad = bc

Your Turn: solve these examples

6

213

x

Ex:Ex:

5

4

2

2

x

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

Scale factor: the ratio of corresponding sides

A

B

C D

EL

M

N O

P

Polygon ABCDE ~ Polygon LMNOP

NO

CD

LM

ABEx:

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

A

B

C

Y

X

Z

ABC ~ XYZ

angle A angle X

angle B angle Y

angle C angle Z

YZ

BC

XZ

AC

XY

AB

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)

A

B

C

R

S

T

1812 6

4

7.3: Similar Triangles

Triangles are similar if you show: All 3 pairs of corresponding sides are

proportional (SSS Similarity)

A

B

C

R

S

T

10

14

6

7

5

3

7.3: Similar Triangles

Triangles are similar if you show: Any 2 pairs of corresponding angles are

congruent (AA Similarity)

A

B

C

R

S

T

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

BC

XY

XB

AX

YC

AY*If XY ll CB, then

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

A

B

CD

E

*If E and B are the midpoints of AD and AC respectively, then EB = DC 2

1

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

EF

DE

BC

AB

AB

C

DE

F

EF

BC

DF

AC

EF

DF

BC

AC

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

BCAB

AB

C

DE

FEFDE If , then

A. In the figure, DE and EF are midsegments of ΔABC. Find AB.B. Find FE.C. Find mAFE.

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 similar, then the perimeters are proportional to the measures of corresponding sides

XZ

AC

YZ

BC

XY

AB

XYZperimeter

ABCperimeter

A

B C

X

Y Z

7.5 : Parts of Similar Triangles

the measures of the corresponding altitudes are proportional to the corresponding sides

the measures of the corresponding angle bisectors are proportional to the corresponding sides

YZ

BC

YX

BA

XZ

AC

XW

AD

A

B C

X

Y Z

D

WL

M

NO

R

S

TU

RT

LN

RS

LM

ST

MN

SU

MO

If two triangles are similar:

7.5 : Parts of Similar Triangles

If 2 triangles are similar, then the measures of the corresponding medians are proportional to the corresponding sides.

An angle bisector in a triangle cuts the opposite side into segments that are proportional to the other sides

G

H IJ

T

U V W

UW

HJ

TW

GJ

UT

GH

TV

GI

A

BC

D

E

FG

H

AD

AB

CD

BC

EH

EF

GH

FG

In the figure, ΔLJK ~ ΔSQR. Find the value of x.

In the figure, ΔABC ~ ΔFGH. Find the value of x.

Find x.

Find n.