Similarity and Dilation: Mapping Figures

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

2. Similarity Module 3 Lesson 5

3. Similarity Informal definition: same shape, not necessarily the same size Notation: ~ B B’ A A’ C’ C

4. 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.

5. 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).

6. Example 2 In 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.

7. 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.

8. Corresponding Sides We 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 L A 16 m 10 m R 7 m 20 m 14 m B T 8 m G

9. Scale Factor The ratio of corresponding side lengths of a figure and its image after dilation. Scale factor = ? L A 16 m 10 m R 7 m 20 m 14 m B T 8 m G

10. Example 3 Are the two triangles similar? 4 cm 7.5 cm 5 cm 6 cm 2 cm 3 cm

11. Example 4 Are the two polygons similar? 8 cm 3 cm 15 cm 6 cm

12. Example 5 Would a dilation map Figure A onto Figure A'? That is, is Figure A ~ Figure A'?

13. 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.

14. Complete your worksheet