EXAMPLE 1 Draw a dilation with a scale factor greater than 1

SOLUTION

Draw a dilation of quadrilateral ABCD with vertices A(2, 1), B(4, 1), C(4, – 1), and D(1, – 1). Use a scale factor of 2.

First draw ABCD. Find the dilation of each vertex by multiplying its coordinates by 2. Then draw the dilation.

D(1, –1) P(2, –2)

B(4, 1) M(8, 2)A(2, 1) L(4, 2)

(x, y) (2x, 2y)

C(4, –1) N(8, –2)

EXAMPLE 2 Verify that a figure is similar to its dilation

a. Sketch ABC and DEF.

b. Verify that ABC and DEF are similar.

A triangle has the vertices A(4,– 4), B(8, 2), and C(8,– 4). The image of ABC after a dilation with a scale factor of is DEF.1

2

EXAMPLE 2 Verify that a figure is similar to its dilation

SOLUTION

A(4, – 4) D(2, – 2)

B(8, 2) E(4, 1)

C(8, – 4) F(4, – 2)

(x, y) 12

12

x, y

The scale factor is less than one, so the dilation is a reduction.

a.

EXAMPLE 2 Verify that a figure is similar to its dilation

ACDF

BCEF=

? 42

63=

So, the lengths of the sides that include C and F are proportional.

Therefore, ABC DEF by the SAS Similarity Theorem.

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ANSWER

Because C and F are both right angles, C F. Show that the lengths of the sides that include C and F are proportional. Find the horizontal and vertical lengths from the coordinate plane.

b.

GUIDED PRACTICE for Examples 1 and 2

1. P(–2, 21), Q(–1, 0), R(0, –1); k = 4

Find the dilation of each vertex by multiplying its coordinates by 4.

P (–2, –1) = L (–8, –4)

Q (–1, 0) = M (– 4, 0)

R (0, –1) = N (0, –4)

ANSWER

SOLUTION

x, y = 4x , 4y

Find the coordinates of L, M, and N so that LMN is a dilation of PQR with a scale factor of k. Sketch PQR and LMN.

GUIDED PRACTICE for Examples 1 and 2

2. P(5, –5), Q(10, –5), R(10, 5); k = 0.4

Find the dilation of each vertex by multiplying its coordinates by 0.4. The scale is less than one, so the dilation is a reduction.

ANSWER

SOLUTION

P(5, –5) L (2, –2)

Q(10,–2) M (4, –2)

R (10, 5) N (4, 2)

x, y

x , y25

25