Question

1. figure $j$ is the image of figure $j$ under a dilation centered a which dilation maps figure $j$ back onto figure $j$?

a. dilation centered at $(0,0)$ with scale factor $k$
b. dilation centered at $(0,0)$ with scale factor $\frac{1}{k}$
c. dilation centered at $(k,k)$ with scale factor $k$
d. dilation centered at $(k,k)$ with scale factor $\frac{1}{k}$
in the diagram, figure $abcd\sim$ figure $fghi$.

Explanation:

Step1: Recall Dilation Properties

Dilation is a transformation that changes the size of a figure. If a dilation with scale factor \( k \) (centered at the same center) maps \( J \) to \( J' \), then to map \( J' \) back to \( J \), we need the inverse dilation. The inverse of a dilation with scale factor \( k \) (centered at the same center) has a scale factor of \( \frac{1}{k} \). Also, the center of dilation must remain the same as the original dilation (since dilation is centered at a point, to reverse it, the center stays the same).

Step2: Analyze the Center and Scale Factor

The original dilation (from \( J \) to \( J' \)) is centered at some point (the problem implies the center for reversing should be the same as the original, but looking at the options, the center for reversing should match the original center. Wait, actually, when you dilate \( J \) to \( J' \) with center \( C \) and scale \( k \), to reverse it, you dilate \( J' \) with center \( C \) and scale \( \frac{1}{k} \). But in the options, let's check:

• Option A: Scale factor \( k \), which would map \( J' \) to a larger/same size, not back to \( J \).
• Option B: Center at \( (0,0) \) (assuming original center was \( (0,0) \)) and scale factor \( \frac{1}{k} \). If original dilation was centered at \( (0,0) \) with scale \( k \), then inverse is \( \frac{1}{k} \) at same center.
• Option C: Center \( (k,k) \) and scale \( k \), wrong center and scale.
• Option D: Center \( (k,k) \), wrong center.

Since dilation center must be the same as original (to reverse the dilation), and scale factor is reciprocal. So the correct dilation is centered at the same center as original (here, likely \( (0,0) \) from options) with scale factor \( \frac{1}{k} \).

Answer:

B. Dilation centered at \( (0, 0) \) with scale factor \( \frac{1}{k} \)