Question

1. circle a is dilated with the origin as the center of dilation to create circle b. ray thinks the scale factor used in the dilation is 1/2. alex thinks the scale factor used in the dilation is 2. who is correct and why?

Explanation:

Step1: Recall the formula for scale factor in dilation

The scale factor \(k\) for a dilation is given by \(k=\frac{\text{radius of image}}{\text{radius of pre - image}}\).
Let the radius of circle \(A\) be \(r_A\) and the radius of circle \(B\) be \(r_B\). From the graph, assume the radius of circle \(A\) (pre - image) is \(4\) units (counting the grid units from the center \((0,0)\) to the edge of circle \(A\)) and the radius of circle \(B\) (image) is \(2\) units.

Step2: Calculate the scale factor

Using the formula \(k = \frac{r_B}{r_A}\), substitute \(r_B = 2\) and \(r_A=4\). Then \(k=\frac{2}{4}=\frac{1}{2}\).

Answer:

Alex is correct. The scale factor \(k\) for a dilation is calculated as \(k=\frac{\text{radius of image}}{\text{radius of pre - image}}\). If the radius of circle \(A\) (pre - image) is \(4\) and the radius of circle \(B\) (image) is \(2\), then \(k = \frac{2}{4}=\frac{1}{2}\)