Holt McDougal Geometry

7-2 Similarity and Transformations7-2 Similarity and Transformations

Holt GeometryHolt McDougal Geometry

Holt McDougal Geometry

7-2 Similarity and Transformations

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)

2. (x, y) → (x, –y); (–2, 1)

3. (x, y) → (3x, 3y); (–5, 0)

4. (x, y) → ; (3, -6) 31 x,

31 y

Holt McDougal Geometry

7-2 Similarity and Transformations

• 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.

• The scale factor describes how the figure is enlarged or reduced.

Holt McDougal Geometry

7-2 Similarity and Transformations

Translations, reflections, and rotations are congruence transformations.

Dilations are similarity transformations.

Remember!

Holt McDougal Geometry

7-2 Similarity and Transformations

D: (x, y) → (3x, 3y) A(1, 1), B(3, 1), C(3, 2)

A. Apply the dilation D to the polygon with the given vertices. Describe the dilation.

Holt McDougal Geometry

7-2 Similarity and Transformations

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 41 x,

41 y

Holt McDougal Geometry

7-2 Similarity and Transformations

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)

Determine whether the polygons with the given vertices are similar. If so, give the rule.

Holt McDougal Geometry

7-2 Similarity and Transformations

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).

Holt McDougal Geometry

7-2 Similarity and Transformations

Explore dilations further

Characteristics Original Triangle ABC

New Triangle A’B’C’

Observations

Coordinates A(3,5) B(3,2) C(7,2)

Angle Measures 53ᵒ - 90ᵒ - 37ᵒ

Length of Sides

Perimeter

Area

Record observations using a scale factor of 2.

Holt McDougal Geometry

7-2 Similarity and Transformations

Characteristics Original Rectangle ABCD

New Rectangle A’B’C’D’

Observations

Coordinates A(-2,4), B(-2,-2), C(4,-2), D (4,4)

Angle Measures 90ᵒ - 90ᵒ -90ᵒ - 90ᵒ

Length of Sides

Perimeter

Area

Record observations using a scale factor of 1/2.Explore dilations further

Holt McDougal Geometry

7-2 Similarity and Transformations

Holt McDougal Geometry

7-2 Similarity and Transformations

A. Circle A with center (0, 0) and radius 1 is similar to circle B with center (0, 6) and radius 3.

Holt McDougal Geometry

7-2 Similarity and Transformations

B. Circle C with center (0, –3) and radius 2 is similar to circle D with center (5, 1) and radius 5.

Holt McDougal Geometry

7-2 Similarity and Transformations

Prove that circle A with center (2, 1) and radius 4 is similar to circle B with center (-1, -1) and radius 2.

Holt McDougal Geometry

7-2 Similarity and Transformations

Characteristics Original Circle with diameter AB

New circle with diameter A’B’

Observations

Coordinates A(4,1) B(4,-3)

Length of diameter 4 cm

Length of radius

Circumference

Area

Record observations using a scale factor of 2Explore dilations further

Holt McDougal Geometry

7-2 Similarity and Transformations

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.

Holt McDougal Geometry

7-2 Similarity and Transformations

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)

Holt McDougal Geometry

7-2 Similarity and Transformations

3. Prove that circle A with center (0, 4) and radius 4 is similar to circle B with center (-2, -7) and radius 6.