Clear everything off your desk except your notebook and pen/pencil.

SimilarityModule 3Lesson 5

SimilarityInformal definition: same shape, not necessarily the same size

Notation:

A’

C’

B’

C

B

A

~

Similarity

Two figures are said to be similar if one can be mapped onto the other using a dilation followed by a congruent transformation (rigid transformations) … or a congruent transformation followed by a dilation.

Knowing that the green triangle and the purple triangle are congruent, discuss with a partner how to describe the

sequence that would map (purple), onto (gray).

Example 2In the diagram below, △ABC~△A'B'C'. Describe the sequence of the dilation followed by a congruence that would prove these figures to be similar.

Example 2One Possible Solution

The sequence that would map △ABC onto △A'B'C' to prove the figures similar is: A dilation from the origin by scale factor , followed by the translation of 4 units down, followed by the reflection across the y-axis.

Corresponding SidesWe can’t use just any two sides to calculate scale factor, we need to look at corresponding sides.

Corresponding sides: sides that are in the same relative position

B

A

T

L

R

G

10 m

20 m8 m

7 m

16 m

14 m

Scale FactorThe ratio of corresponding side lengths of a figure and its image after dilation.

Scale factor = ?

B

A

T

L

R

G

10 m

20 m8 m

7 m

16 m

14 m

Example 3

Are the two triangles similar?

6 cm

2 cm

4 cm

3 cm

7.5 cm

5 cm

Example 4

Are the two polygons similar?

8 cm

6 cm

3 cm 15 cm

Example 5Would a dilation map Figure A onto Figure A'?

That is, is Figure A ~ Figure A'?

Example 5 Solution

No. Even though two sets of sides are in proportion, there exists no single rigid motion or sequence of rigid motions that would map a four-sided figure to a three-sided figure.

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