1. Pearson Unit 3 Topic 9: Similarity 9-2: Similarity Transformations Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

2. TEKS Focus:
(7)(A) Apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles.
(1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
(1)(E) Create and use representations to organize, record, and communicate mathematical ideas.
(1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.
(3)(A) Describe and perform transformations of figures in a plane using coordinate notation.
(3)(B) Determine the image or pre-image of a given two-dimensional figures under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane.
(3)(C) Identify the sequence of transformations that will carry a given pre-image onto an image on and off the coordinate plane.
(5)(C) Use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships.

3. A dilation is a transformation that changes the size of a figure but not its shape. The preimage and the image are always similar.
A scale factor describes how much the figure is enlarged or reduced.
For a dilation with scale factor k, you can find the image of a point by multiplying each coordinate by k: (a, b)  (ka, kb).

4. If the scale factor of a dilation is greater than 1 (k > 1), it is an enlargement. The scale factor in fraction form is an improper fraction.
If the scale factor is less than 1 (k < 1), it is a reduction. The scale factor in fraction form is a proper fraction.
Helpful Hint
0 < k < 1,

5. Example 1:

6. Example: 2
Draw the border of the photo after a dilation with scale factor

7. Example: 2 continued
Step 1 Multiply the vertices of the photo A(0, 0), B(0, 4), C(3, 4), and D(3, 0) by
Rectangle
ABCD
Rectangle
A’B’C’D’

8. Example: 2 continued
Step 2 Plot points A’(0, 0), B’(0, 10), C’(7.5, 10), and D’(7.5, 0).
Draw the rectangle—you will need to extend the axes on your
notes to graph this rectangle.

9. Example 3: T 4, 2 L’ (0, 4) M’ (1, -1) N’ (3, 3) D 0.5 L’’ (0, 2)

10. Example: 4
Given that ∆TUO ~ ∆RSO, find the coordinates of U and the scale factor.
Remember that distance is always positive! Since ∆TUO ~ ∆RSO,
9 = OU
12(OU) = 144
OU = 12
Coordinates of U(0, 12).
If this is a reduction, the scale
factor is 9/12 = ¾.
If this is an enlargement, the
scale factor is 12/9 = 4/3.

11. Example: 5 𝑅 𝑦−𝑎𝑥𝑖𝑠 (ABCD) ° 𝐷 .5 (ABCD) A(-6, 2) B(-2, 2) C(-1, -2)
N(1, 1)
H(.5, -1)
P(4, -1)
𝑅 𝑦−𝑎𝑥𝑖𝑠 (ABCD) ° 𝐷 .5 (ABCD)

12. Example: 6 Given: R(–2, 0), S(–3, 1), T(0, 1), U(–5, 3), and V(4, 3).
Prove: ∆RST ~ ∆RUV
Step 1 Plot the points and draw the triangles. R S T U V

13. Example: 6 continued
R(–2, 0), S(–3, 1), T(0, 1), U(–5, 3), and V(4, 3).
Step 2 Use the Distance Formula to find the side lengths. R S T U V

14. Example: 6 continued 18 2 = 9 2 2 = 3 2 2 = 3 45 5 = 9 5 5 = 3 5 5 = 3
Step 3 Prove the definition of similarity:
∡R  ∡R by Reflexive Property of Congrunce
∡U  ∡TSR by Corresponding Angles Postulate
∡V  ∡STR by Corresponding Angles Postulate
Compare all of the corresponding sides:
= = = 3
= = = 3
9 3 = 3
Therefore ∆RST ~ ∆RUV .

15. Example: 7 Graph the image of ∆ABC after a dilation with scale factor
Verify that ∆A'B'C' ~ ∆ABC.

16. Example: 7 continued
Step 1 Multiply each coordinate by to find the coordinates of the vertices of ∆A’B’C’.

17. Example: 7 continued B’ (2, 4) A’ (0, 2) C’ (4, 0)
Step 2 Graph ∆A’B’C’.
B’ (2, 4)
A’ (0, 2)
C’ (4, 0)

18. Example: 7 continued
Step 3 Use the Distance Formula to find the side lengths.

19. Example: 7 continued ∆A’B’C’ ~ ∆ABC .
Step 4 Find the similarity ratio.
Since , and all corresponding angles are congruent.
∆A’B’C’ ~ ∆ABC .

20. Example: 8
Graph the image of ∆MNP after a dilation with scale factor 3.
Verify that ∆M 'N 'P ' ~ ∆MNP.

21. Example: 8 continued
Step 1 Multiply each coordinate by 3 to find the coordinates of the vertices of ∆M’N’P’.

22. Example: 8 continued
Step 2 Graph ∆M’N’P’.

23. Example: 8 continued
Step 3 Use the Distance Formula to find the side lengths.

24. Example: 8 continued S→ S→ S→ ∆MNP ~ ∆M’N’P’ .
Step 4 Find the similarity ratio. S→ S→ S→
Since , and all corresponding angles are congruent.
∆MNP ~ ∆M’N’P’ .

25. Example: 9
10  12  16
∆JKL ~ TSR

26. Example: 10 2x + 4 = 3x -3 7 = x ML = LN = 18 MN = 28 12 = 12 = 18 2/3
12 = 12 = /3
2/3 = 2/3 = 2/3
Perimeters: 42 2/3 = 2