Question

figure ( j ) is the image of figure ( j ) under a dilation centered at ( (0,0) ) with scale factor ( k ). which dilation maps figure ( j ) back onto figure ( j )?
dilation centered at ( (0,0) ) with scale factor ( k )
dilation centered at ( (0,0) ) with scale factor ( \frac{1}{k} )
dilation centered at ( (k,k) ) with scale factor ( k )
dilation centered at ( (k,k) ) with scale factor ( \frac{1}{k} )

Explanation:

Step1: Recall the property of dilation

If a figure \(J\) is dilated with scale factor \(k\) centered at \((0,0)\) to get \(J'\), then to map \(J'\) back to \(J\), we use the inverse operation.

Step2: Determine the inverse scale factor

The inverse of a scale factor \(k\) is \(\frac{1}{k}\), and the center of dilation remains the same \((0,0)\) since the original dilation was centered at \((0,0)\).

Answer:

Dilation centered at \((0,0)\) with scale factor \(\frac{1}{k}\) (the second option in the given list).