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Holt Geometry
12-7 Dilations12-7 Dilations
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Geometry
12-7 Dilations
Warm Up
1. Translate the triangle with vertices A(2, –1), B(4, 3), and C(–5, 4) along the vector <2, 2>.
2. ∆ABC ~ ∆JKL. Find the value of JK.
A'(4,1), B'(6, 5),C(–3, 6)
Holt Geometry
12-7 Dilations
Identify and draw dilations.
Objective
Holt Geometry
12-7 Dilations
center of dilationenlargementreduction
Vocabulary
Holt Geometry
12-7 Dilations
Recall that a dilation is a transformation that changes the size of a figure but not the shape. The image and the preimage of a figure under a dilation are similar.
Holt Geometry
12-7 Dilations
Example 1: Identifying Dilations
Tell whether each transformation appears to be a dilation. Explain.
A. B.
Yes; the figures are similar and the image is not turned or flipped.
No; the figures are not similar.
Holt Geometry
12-7 Dilations
Check It Out! Example 1
a. b.
Yes, the figures are similar and the image is not turned or flipped.
No, the figures are not similar.
Tell whether each transformation appears to be a dilation. Explain.
Holt Geometry
12-7 Dilations
For a dilation with scale factor k, if k > 0, the figure is not turned or flipped. If k < 0, the figure is rotated by 180°.
Helpful Hint
Holt Geometry
12-7 Dilations
Holt Geometry
12-7 Dilations
A dilation enlarges or reduces all dimensions proportionally. A dilation with a scale factor greater than 1 is an enlargement, or expansion. A dilation with a scale factor greater than 0 but less than 1 is a reduction, or contraction.
Holt Geometry
12-7 Dilations
Example 2: Drawing Dilations
Copy the figure and the center of dilation P. Draw the image of ∆WXYZ under a dilation with a scale factor of 2.
Step 1 Draw a line through P and each vertex.
Step 2 On each line, mark twice the distance from P to the vertex.
Step 3 Connect the vertices of the image.
W’ X’
Z’Y’
Holt Geometry
12-7 Dilations
Check It Out! Example 2
Copy the figure and the center of dilation. Draw the dilation of RSTU using center Q and a scale factor of 3.
Step 1 Draw a line through Q and each vertex.
Step 2 On each line, mark twice the distance from Q to the vertex.
Step 3 Connect the vertices of the image.
R’ S’
T’U’
Holt Geometry
12-7 Dilations
Example 3: Drawing Dilations
On a sketch of a flower, 4 in. represent 1 in. on the actual flower. If the flower has a 3 in. diameter in the sketch, find the diameter of the actual flower.
The scale factor in the dilation is 4, so a 1 in. by 1 in. square of the actual flower is represented by a 4 in. by 4 in. square on the sketch.
Let the actual diameter of the flower be d in.
3 = 4d
d = 0.75 in.
Holt Geometry
12-7 Dilations
Check It Out! Example 3
What if…? An artist is creating a large painting from a photograph into square and dilating each square by a factor of 4. Suppose the photograph is a square with sides of length 10 in. Find the area of the painting.
The scale factor of the dilation is 4, so a 10 in. by 10 in. square on the photograph represents a 40 in. by 40 in. square on the painting.
Find the area of the painting.
A = l w = 4(10) 4(10)
= 40 40 = 1600 in2
Holt Geometry
12-7 Dilations
Holt Geometry
12-7 Dilations
If the scale factor of a dilation is negative, the preimage is rotated by 180°. For k > 0, a dilation with a scale factor of –k is equivalent to the composition of a dilation with a scale factor of k that is rotated 180° about the center of dilation.
Holt Geometry
12-7 Dilations
Example 4: Drawing Dilations in the Coordinate Plane
Draw the image of the triangle with vertices P(–4, 4), Q(–2, –2), and R(4, 0) under a
dilation with a scale factor of centered at the origin.
The dilation of (x, y) is
Holt Geometry
12-7 Dilations
Example 4 Continued
Graph the preimage and image.
P’
Q’R’
P
R
Q
Holt Geometry
12-7 Dilations
Check It Out! Example 4 Draw the image of the triangle with vertices R(0, 0), S(4, 0), T(2, -2), and U(–2, –2) under a dilation centered at the origin with a scale factor of .
The dilation of (x, y) is
Holt Geometry
12-7 Dilations
Check It Out! Example 4 Continued
Graph the preimage and image.
RS
TU
R’
S’T’ U’
Holt Geometry
12-7 Dilations
Lesson Quiz: Part I
1. Tell whether the transformation appears to be a dilation.
yes
2. Copy ∆RST and the center of dilation. Draw the image of ∆RST under a dilation with a scale of .
Holt Geometry
12-7 Dilations
3. A rectangle on a transparency has length 6cm and width 4 cm and with 4 cm. On the transparency 1 cm represents 12 cm on the projection. Find the perimeter of the rectangle in the projection.
Lesson Quiz: Part II
4. Draw the image of the triangle with vertices E(2, 1), F(1, 2), and G(–2, 2) under a dilation with a scale factor of –2 centered at the origin.
240 cm