Question

each coordinate plane below shows a preimage and its dilated image. all dilations are centered at the origin. write a sentence to describe each dilation. the first problem has been completed for you.
1
triangle efg was dilated by a scale factor of \\(\frac{1}{2}\\) to form triangle efg.
2
(blank lines for answer)
3
(blank lines for answer)
4
(blank lines for answer)

Explanation:

Step1: Analyze Problem 2

First, identify the preimage and image. Preimage: N, O, P, Q. Image: N', O', P', Q'. Check coordinates. Let's find a vertex, e.g., N: (-8, 8), N': (-2, 2). Scale factor: \( \frac{2}{8} = \frac{1}{4} \)? Wait, no: N' is (-2,2), N is (-8,8). Wait, \( \frac{-2}{-8} = \frac{1}{4} \), \( \frac{2}{8} = \frac{1}{4} \). Wait, or maybe preimage is N', O', P', Q' and image is N, O, P, Q? No, the image is smaller? Wait, N is at (-8,8), N' at (-2,2). So from N to N', the scale factor is \( \frac{1}{4} \)? Wait, no: dilation center at origin, so image = preimage * scale factor. So if preimage is N(-8,8), image N'(-2,2), then scale factor \( k \) is \( \frac{-2}{-8} = \frac{1}{4} \), \( \frac{2}{8} = \frac{1}{4} \). Wait, but the image is smaller, so scale factor \( \frac{1}{4} \)? Wait, no, maybe I got preimage and image reversed. Wait, the problem says "preimage and its dilated image". So preimage is the original, image is dilated. So in problem 2, the larger figure is N, O, P, Q (preimage), smaller is N', O', P', Q' (image). So scale factor is \( \frac{1}{4} \)? Wait, N is (-8,8), N' is (-2,2). So \( (-8) \times k = -2 \implies k = \frac{1}{4} \), \( 8 \times k = 2 \implies k = \frac{1}{4} \). So the dilation is with scale factor \( \frac{1}{4} \).

Step2: Describe Problem 2

So the preimage is quadrilateral NOPQ, dilated by scale factor \( \frac{1}{4} \) centered at origin to form quadrilateral N'O'P'Q'.

Step3: Analyze Problem 3

Preimage: J, K, L. Image: J', K', L'. Let's take J: (-2,1), J': (-6,3). Scale factor: \( \frac{-6}{-2} = 3 \), \( \frac{3}{1} = 3 \). K: (2,2), K': (6,6). \( \frac{6}{2}=3 \), \( \frac{6}{2}=3 \). L: (-2,-3), L': (-6,-9). \( \frac{-6}{-2}=3 \), \( \frac{-9}{-3}=3 \). So scale factor 3. So triangle JKL (preimage) dilated by scale factor 3 to form triangle J'K'L'.

Step4: Analyze Problem 4

Preimage: W, X, Y, Z. Image: W', X', Y', Z'. Let's take W: (1,3), W': (0,6). Wait, no: W is (1,3), W' is (0,6)? Wait, coordinates: W(1,3), W'(0,6)? Wait, Z: (-1,-1), Z'(-1,-2)? Wait, maybe better: W(1,3), W'(0,6)? No, let's check Z: Z(-1,-1), Z'(-1,-2)? Wait, no, looking at the graph: W is (1,3), W' is (0,6)? Wait, maybe preimage is W, X, Y, Z (smaller), image W', X', Y', Z' (larger). Let's take Z: (-1,-1), Z'(-1,-2)? No, Z' is at (-1,-2)? Wait, no, Z is at (-1,-1), Z' at (-1,-2)? No, the distance from origin: Z(-1,-1), Z'(-1,-2). Wait, no, maybe scale factor 2. W(1,3), W'(0,6)? No, W is (1,3), W' is (0,6)? Wait, X(2,3), X'(4,6). Y(3,-1), Y'(6,-2). Z(-1,-1), Z'(-2,-2). Ah! So X(2,3) to X'(4,6): scale factor 2. Y(3,-1) to Y'(6,-2): scale factor 2. Z(-1,-1) to Z'(-2,-2): scale factor 2. W(1,3) to W'(0,6)? No, W is (1,3), W' is (0,6)? Wait, no, W is (1,3), W' is (0,6)? Wait, maybe W is (0,3)? Wait, the graph: W is at (1,3)? Wait, maybe I misread. Let's see: Z is at (-1,-1), Z' at (-2,-2). Y is at (3,-1), Y' at (6,-2). X is at (2,3), X' at (4,6). W is at (0,3), W' at (0,6). Yes! So W(0,3) to W'(0,6): scale factor 2. X(2,3) to X'(4,6): scale factor 2. Y(3,-1) to Y'(6,-2): scale factor 2. Z(-1,-1) to Z'(-2,-2): scale factor 2. So scale factor 2. So trapezoid WXYZ (preimage) dilated by scale factor 2 centered at origin to form trapezoid W'X'Y'Z'.

Answer:

s (for each problem):

Problem 2:

Quadrilateral NOPQ was dilated by a scale factor of \( \frac{1}{4} \) centered at the origin to form quadrilateral N'O'P'Q'.

Problem 3:

Triangle JKL was dilated by a scale factor of 3 centered at the origin to form triangle J'K'L'.

Problem 4:

Trapezoid WXYZ was dilated by a scale factor of 2 centered at the origin to form trapezoid W'X'Y'Z'.