Guggenheim MuseumGuggenheim Museum

Building big stuff can be expensive. So to work out details, artists and architects usually build scale models.

Guggenheim MuseumGuggenheim Museum

A scale model is similarsimilar to the actually object that is to be built. And that does not mean that they are kind of alike.

Guggenheim MuseumGuggenheim Museum

A scale model is similarsimilar to the actually object that is to be built. And that does not mean that they are kind of alike.

SimilaritySimilarity

Figures that have the same shape but not necessarily the same size are similar figuressimilar figures. But what does “same shape mean”? Are the triangles similar?

NOT Similar

SimilaritySimilarity

Similar shapes can be thought of as enlargementsenlargements or reductionsreductions with no irregular distortions.– So two shapes are similar if

one can be enlarged or reduced so that it is congruent to the original.

6.3: Use Similar Polygons6.3: Use Similar Polygons

Objectives:

1. To define similar polygons

2. To find missing measures in similar polygons

3. To find the perimeter of similar polygons using a scale factor

Similar PolygonsSimilar Polygons

Two polygons are similar polygonssimilar polygons if the corresponding angles are congruent and the corresponding sides are proportional.

MAIZCORN ~

ZMNC

IZRN

AIOR

MACO

ZNIR

AOMC

C

OR

N

C

OR

NM

A

I

Z

Similarity Statement:Similarity Statement:

Corresponding Angles:Corresponding Angles:

Statement of Proportionality:Statement of Proportionality:

Example 1Example 1

Use the definition of similar polygons to find the measure of x and y, assuming SMAL ~ BIGE.

D

E

F

A

B

C

6

3

5

8

10

Example 2Example 2

When asked to find the length of segment DE given that the triangles are similar, Kenny says 10. Explain what is wrong with Kenny’s reasoning?

Example 3Example 3

Determine whether or not the polygons below are similar.

Scale FactorScale Factor

In similar polygons, the ratio of two corresponding sides is called a scale factorscale factor.

What is the scale factor of the similar polygons shown?

C

OR

N

M

A

I

Z

4

8

5

6

6

12

9

7.5

Scale FactorScale Factor

Explain why the scale factor will always be the same for any two corresponding sides.

C

OR

N

M

A

I

Z

4

8

5

6

6

12

9

7.5

Example 4Example 4

An artist painted a mural from the photograph shown at the right.

If the artist used a scale of ½ inch to represent 1 foot, what best represents the dimensions in feet of the mural?

Example 5Example 5

A. , because corresponding angles of similar triangles are congruent.

B. MK/MN = KJ/NL, because the ratios of the lengths of corresponding sides of similar triangles are equal.

If , which of the following must be true?

JKM NLM

~MKJ MNL

Example 5Example 5

C. KJ/LN = ML/MK, because the ratios of the lengths of corresponding sides of similar triangles are equal.

D. , because corresponding angles of similar triangles are congruent.

If , which of the following must be true?

~MKJ MNL KJM MNL

Example 6Example 6

In the diagrams shown, CORN~MAIZ. Recall that the scale factor of MAIZ to CORN is 3/2 or 1.5. Find the perimeter of each figure. What is the ratio of the perimeter of MAIZ to CORN?

C

OR

N

M

A

I

Z

4

8

5

6

6

12

9

7.5

Perimeter of Similar Perimeter of Similar PolygonsPolygonsIf two polygons are similar, then the ratio of

their perimeters is equal to the ratios of their corresponding side lengths.

Example 7Example 7

In the diagram, ABCDE ~ FGHJK. Find the perimeter of ABCDE.

A

E D

C

B

F

K J

H

G

10

15

9

12

15

18

Example 8Example 8

The polygons below are congruent. Are they also similar? If so, what is the scale factor?

Corresponding LengthsCorresponding Lengths

Corresponding Lengths in Similar Corresponding Lengths in Similar PolygonsPolygons

If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons.

Sides Altitudes

Medians Midsegments

Example 9Example 9

In the diagram ΔTPR ~ ΔXPZ. Find the length of the altitude PS.

AssignmentAssignment

• P. 367-9: 11, 13, 14, 22, 23

• P. 376-8: 1-3, 6, 8-13, 19, 20, 31, 32, 36, 39, 40-42

• Challenge Problems