Similarity and
and Transformations
Transformations
Similarity
Warm Up
Lesson Presentation
Lesson Quiz
Holt
HoltGeometry
McDougal Geometry
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)
4. (x, y) → 1 x, 1 y ; (3, -6)
3
3
(1,-2)
Holt McDougal Geometry
Similarity and Transformations
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.
Holt McDougal Geometry
Similarity and Transformations
Vocabulary
similarity transformation
Holt McDougal Geometry
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.
Holt McDougal Geometry
Similarity and Transformations
Remember!
Translations, reflections, and rotations are
congruence transformations.
Holt McDougal Geometry
Similarity and Transformations
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
Holt McDougal Geometry
Similarity and Transformations
Example 1: Continued
B. Apply the dilation D to the polygon with the
given vertices. Describe the dilation.
D: (x, y) → 3 x, 3 y
4
4
P(–8, 4), Q(–4, 8), R(4, 4)
dilation with center (0, 0) and scale factor 3
4
Holt McDougal Geometry
Similarity and Transformations
Check It Out! Example 1
Apply the dilation D : (x, y)→ 1 x, 1 y to the
4
4
polygon with vertices D(-8, 0), E(-8, -4), and F(4, -8). Name the coordinates of the image points.
Describe the dilation.
D'(-2, 0), E'(-2, -1), F'(-1, -2); dilation with center
1
(0, 0) and scale factor 4
Holt McDougal Geometry
Similarity and Transformations
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:
1x,1y
(x, y) →
3
3
Holt McDougal Geometry
Similarity and Transformations
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.
Holt McDougal Geometry
Similarity and Transformations
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).
Holt McDougal Geometry
Similarity and Transformations
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 x , 1 y
3
3
Holt McDougal Geometry
Similarity and Transformations
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).
Holt McDougal Geometry
Similarity and Transformations
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.
Holt McDougal Geometry
Similarity and Transformations
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. Circle C with center (0, –3) and radius 2 is
similar to circle D with center (5, 1) and radius
5.
Holt McDougal Geometry
Similarity and Transformations
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.
Holt McDougal Geometry
Similarity and Transformations
Check It Out! Example 3
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 1 So, circles A and B are
2
similar.
Holt McDougal Geometry
Similarity and Transformations
Example 4: Application
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?
Holt McDougal Geometry
Similarity and Transformations
Example 4: Continue
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).
Holt McDougal Geometry
Similarity and Transformations
Check It Out! Example 4
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).
Holt McDougal Geometry
Similarity and Transformations
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
3
(0, 0) and scale factor 2
Holt McDougal Geometry
Similarity and Transformations
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).
Holt McDougal Geometry
Similarity and Transformations
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 So circle A and circle B are similar.
2
Holt McDougal Geometry