Similarity and Transformations - Neshaminy School …€¦ · · 2016-03-08Holt Geometry Warm Up...

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Holt McDougal Geometry

7-2 Similarity and Transformations 7-2 Similarity and Transformations

Holt Geometry

Warm Up

Lesson Presentation

Lesson Quiz

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) (5, 3)

2. (x, y) → (x, –y); (–2, 1) (-2,-1)

3. (x, y) → (3x, 3y); (–5, 0) (-15,-0)

(1,-2) 4. (x, y) → ; (3, -6) 3 1 x,

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.

Objectives

similarity transformation

Vocabulary

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.

Translations, reflections, and rotations are congruence transformations.

Remember!

Example 1: Drawing and Describing Dilations

D: (x, y) → (3x, 3y)

A(1, 1), B(3, 1), C(3, 2)

dilation with center (0, 0) and scale factor 3

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

Example 1: Continued

dilation with center (0, 0) and scale factor 3 4

D: (x, y) →

P(–8, 4), Q(–4, 8), R(4, 4)

4 3 x,

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

Check It Out! Example 1

D'(-2, 0), E'(-2, -1), F'(-1, -2); dilation with center

(0, 0) and scale factor 1 4

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,

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)

Example 2 : Determining Whether Polygons are

Similar

Yes; ABCD maps to HJKL by a dilation:

(x, y) → 1

Determine whether the polygons with the given vertices are similar.

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.

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).

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

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).

Example 3: Proving Circles Similar

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

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

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.

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

Circle A can be mapped to circle A' by a

translation: (x, y) → (x - 3, y - 2). Then circle A'

can be mapped to circle B by a dilation with center

(-1, -1) and scale factor So, circles A and B are

similar.

Eric wants to make a drawing to show how he will arrange three rugs in his room.

The large rug is twice the size of each small rug. Eric will first draw the small rug in the lower left corner and then the large rug in the middle. How can he draw those rugs?

Example 4: Application

Place the lower left rug in the first quadrant of a coordinate plane in a convenient position. Apply the dilation with center (0, 0) and scale factor 2: (x, y) → (2x, 2y).

Example 4: Continue

What if…? How could Tia draw the middle flag to make it 4 times the size of each of the other flags?

Apply the dilation with center (0, 0) and scale

factor 4: (x, y) → (4x, 4y).

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

Lesson Quiz : Part-II

No; (x, y) → (0.5x, 0.5y) maps A to P, but not B to Q.

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)

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).

Lesson Quiz : Part-III

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

Circle A can be mapped to circle A’ by a translation:

(x, y) → (x - 2, y - 11). Circle A’ and circle B both

have center (-2, -7). Then circle A’ can be mapped to

circle B by a dilation with center (-2, -7) and scale

factor 3 2

So circle A and circle B are similar.

Similarity and Transformations - Neshaminy School …€¦ · · 2016-03-08Holt Geometry Warm Up...

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