1. Similarity and Transformations
7-2
Similarity and Transformations
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry

2. Warm Up
Find the image point when the indicated transformation is applied to the given pre-image point.
1. (x, y) → (x + 3, y – 1); (2, 4)
(5, 3)
2. (x, y) → (x, –y); (–2, 1)
(-2,-1)
3. (x, y) → (3x, 3y); (–5, 0)
(-15, 0) 3 1 x, y
4. (x, y) →
; (3, -6)
(1,-2)

3. Objectives
Draw and describe similarity transformations in the coordinate plane.
Use properties of similarity transformations to determine whether polygons are similar and to prove circles similar.

4. Vocabulary
similarity transformation

5. A transformation that produces similar figures is a similarity transformation. A similarity transformation is a dilation or a composite of one or more dilations and one or more congruence transformations. Two figures are similar if and only if there is a similarity transformation that maps one figure to the other figure.

6. Translations, reflections, and rotations are congruence transformations.
Remember!

7. Example 1: Drawing and Describing Dilations
A. Apply the dilation D to the polygon with the given vertices. Describe the dilation.
D: (x, y) → (3x, 3y) A(1, 1), B(3, 1), C(3, 2)
dilation with center (0, 0) and scale factor 3

8. Example 1: Continued
B. Apply the dilation D to the polygon with the given vertices. Describe the dilation.
D: (x, y) →
P(–8, 4), Q(–4, 8), R(4, 4) 4 3 x, y
dilation with center (0, 0) and scale factor 3 4

9. Check It Out! Example 1
Apply the dilation D : (x, y)→
polygon with vertices D(-8, 0), E(-8, -4), and F(-4, -8). Name the coordinates of the image points. Describe the dilation.
to the 4 1 x, y
D'(-2, 0), E'(-2, -1), F'(-1, -2); dilation with center
(0, 0) and scale factor 1 4

10. Example 2 : Determining Whether Polygons are Similar
Determine whether the polygons with the given vertices are similar.
A. A(–6, -6), B(-6, 3), C(3, 3),
D(3, -6) and H(-2, -2),
J(-2, 1), K(1, 1), L(1, -2)
Yes; ABCD maps to HJKL by a dilation:
(x, y) → 1 3 x y ,

11. Example 2: Continued
B. P(2, 0), Q(2, 4), R(4, 4),
S(4, 0) and W(5, 0),
X(5, 10), Y(8, 10), Z(8, 0).
No; (x, y) → (2.5x, 2.5y) maps P to W, but not S to Z.

12. Example 2: Continued
C. A(1, 2), B(2, 2), C(1, 4) and
D(4, -6), E(6, -6), F(4, -2)
Yes; ABC maps to A’B’C’ by a translation: (x, y) → (x + 1, y - 5). Then A’B’C’ maps to DEF by a dilation: (x, y) → (2x, 2y).

13. Example 2: Continued
D. F(3, 3), G(3, 6), H(9, 3),
J(9, –3) and S(–1, 1),
T(–1, 2), U(–3, 1), V(–3, –1).
Yes; FGHJ maps to F’G’H’J’ by a reflection : (x, y) → (-x, y). Then F’G’H’J’ maps to STUV by a dilation:
(x, y) 1 3 x y ,

14. Check It Out! Example 2
Determine whether the polygons with the given vertices are similar : A(2, -1), B(3, -1), C(3, -4) and P(3, 6), Q(3, 9), R(12, 9).
The triangles are similar because ABC can be mapped to A'B'C' by a rotation: (x, y) → (-y, x), and then A'B'C' can be mapped to PQR by a dilation: (x, y) → (3x, 3y).

15. Example 3: Proving Circles Similar
A. Prove that Circle A with center (0, 0) and radius 1 is similar to circle B with center (0, 6) and radius 3.

16. Example 3: Continued
Circle A can be mapped to circle A’ by a translation: (x, y) → (x, y + 6). Circle A’ and circle B both have center (0, 6). Then circle A’ can be mapped to circle B by a dilation with center (0, 6) and scale factor 3. So circle A and circle B are similar.
B. Prove that Circle C with center (0, –3) and radius 2 is similar to circle D with center (5, 1) and radius 5.

17. Example 3: Continued
Circle C can be mapped to circle C’ by a translation: (x, y) → (x + 5, y + 4). Circle C’ and circle D both have center (5, 1).Then circle C’ can be mapped to circle D by a dilation with center (5, 1) and scale factor 2.5. So circle C and circle D are similar.

18. Lesson Quiz : Part-I
1. Apply the dilation D: (x, y)
to the
polygon with vertices A(2, 4), B(2, 6), and C(6, 4). Name the coordinates of the image points. Describe the dilation.
A’(3, 6), B’(3, 9), C’(9, 6); dilation with center
(0, 0) and scale factor 3 2

19. Lesson Quiz : Part-II
Determine whether the polygons with the given vertices are similar.
2. A(-4, 4), B(6, 4), C(6, -4),
D(-4, -4) and P(-2, 2),
Q(4, 2), R(4, -2), S(-2, -2)
No; (x, y) → (0.5x, 0.5y) maps A to P, but not B to Q.
3. A(2, 2), B(2, 4), C(6, 4) and
D(3, -3), E(3, -6), F(9, -6)
Yes; △ ABC maps to △ A’B’C’ by a reflection: (x, y) → (x, -y). Then △ A’B’C’ maps to △DEF by a dilation:(x, y) → (1.5x, 1.5y).