EXAMPLE 1 Draw a dilation with a scale factor greater than 1 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. SOLUTION First draw ABCD. Find the dilation of each vertex by multiplying its coordinates by 2. Then draw the dilation. (x, y) (2x, 2y) A(2, 1) L(4, 2) B(4, 1) M(8, 2) C(4, –1) N(8, –2) D(1, –1) P(2, –2)

EXAMPLE 2 Verify that a figure is similar to its dilation 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 a. Sketch ABC and DEF. b. Verify that ABC and DEF are similar.

EXAMPLE 2 Verify that a figure is similar to its dilation SOLUTION The scale factor is less than one, so the dilation is a reduction. a. (x, y) 1 2 x, y A(4, – 4) D(2, – 2) B(8, 2) E(4, 1) C(8, – 4) F(4, – 2)

Verify that a figure is similar to its dilation EXAMPLE 2 Verify that a figure is similar to its dilation 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. AC DF BC EF = ? 4 2 6 3 = So, the lengths of the sides that include C and F are proportional. Therefore, ABC DEF by the SAS Similarity Theorem. ~ ANSWER

GUIDED PRACTICE for Examples 1 and 2 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. 1. P(–2, 21), Q(–1, 0), R(0, –1); k = 4 SOLUTION Find the dilation of each vertex by multiplying its coordinates by 4. ANSWER x, y = 4x , 4y P (–2, –1) = L (–8, –4) Q (–1, 0) = M (– 4, 0) R (0, –1) = N (0, –4)

GUIDED PRACTICE for Examples 1 and 2 2. P(5, –5), Q(10, –5), R(10, 5); k = 0.4 SOLUTION 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 x, y x , y 2 5 P(5, –5) L (2, –2) Q(10,–2) M (4, –2) R (10, 5) N (4, 2)