Guggenheim Museum

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

Guggenheim Museum

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

Guggenheim Museum

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

Similarity

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

Similarity

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

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

- It’s like you’ve zoomed in or out on the picture

6.3: 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 Polygons

Two polygons are similar polygons iff 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:

Corresponding Angles:

Statement of Proportionality:

MAKE SURE the parts match up in your statements!!!

Example 1

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

x=28

y=83

D

E

F

A

B

C

6

3

5

8

10

Example 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?

Answer in your notebook

Example 3

Determine whether or not the polygons below are similar.

No. Explain why not in your notebook

Scale Factor

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

2/3 OR 3/2

Scale 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

7.5Answer in your notebook.

Example 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?

6 ft. x 10 ft.

Example 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 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 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

Answer in your notebook.

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

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

Example 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

103.5

Example 8

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

Answer in your notebook.

Corresponding Lengths

Corresponding Lengths in Similar Polygons

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 9

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

15