Question

triangle xyz was dilated using the rule $d_{o,0.25}(x, y) \to (0.25x, 0.25y)$. the image is shown in the diagram. what are the coordinates of z of the pre - image? \
options: $(-2, -1)$, $(-0.25, -0.5)$, $(-8, -4)$, $(-8, 2)$

Explanation:

Step1: Identify Image Coordinates of Z

From the diagram, find the coordinates of \( Z' \) (image of \( Z \)). Assume \( Z' \) has coordinates \( (x', y') \). Looking at the grid, \( Z' \) is at \( (-2, -1) \)? Wait, no—wait, the dilation rule is \( D_{O, 0.25}(x, y) \to (0.25x, 0.25y) \), so image \( (x', y') = (0.25x, 0.25y) \), where \( (x, y) \) is pre - image. Wait, no: dilation with scale factor \( k = 0.25 \) means pre - image \( (x,y) \) maps to image \( (kx, ky) \). So to find pre - image, we solve \( x'=0.25x \) and \( y' = 0.25y \) for \( x \) and \( y \). So \( x=\frac{x'}{0.25}=4x' \), \( y = \frac{y'}{0.25}=4y' \).

First, find \( Z' \) coordinates. From the graph, let's look at the triangle. Let's assume \( Z' \) is at \( (-2, -1) \)? Wait, no, maybe I misread. Wait, the image triangle: let's check the grid. Let's say \( Z' \) has coordinates \( (-2, -1) \)? Wait, no, let's re - examine. Wait, the dilation rule is \( (x,y)\to(0.25x, 0.25y) \), so image is \( (0.25x, 0.25y) \). So pre - image \( (x,y)=( \frac{x'}{0.25},\frac{y'}{0.25})=(4x',4y') \).

Wait, maybe the image \( Z' \) is at \( (-2, -1) \)? No, wait, let's look at the options. Wait, the options for pre - image are \( (-8, -4) \), \( (-2, -1) \), \( (-0.25, -0.5) \), \( (-8, 2) \). Wait, maybe \( Z' \) is at \( (-2, -1) \)? No, wait, if pre - image is \( (-8, -4) \), then image would be \( (0.25\times(-8),0.25\times(-4))=(-2, -1) \), which matches the image. Wait, let's do it properly. Let image \( Z'=(x', y') \). Let's say from the diagram, \( Z' \) is \( (-2, -1) \). Then pre - image \( Z=(x,y) \) satisfies \( 0.25x=-2 \) and \( 0.25y=-1 \). Solving for \( x \): \( x=\frac{-2}{0.25}=-8 \), \( y=\frac{-1}{0.25}=-4 \). So pre - image \( Z \) is \( (-8, -4) \).

Step2: Verify the Calculation

We know that dilation with scale factor \( k = 0.25 \) means \( (x,y)\to(kx,ky) \). So if pre - image is \( (x,y)=(-8, -4) \), then image is \( (0.25\times(-8),0.25\times(-4))=(-2, -1) \), which should match the image coordinates of \( Z \) from the diagram. So this checks out.

Answer:

\((-8, -4)\)