Question

line segment su is dilated to create su using point q as the center of dilation. the scale factor of the dilation is \boxed{}.

Explanation:

Step1: Recall dilation scale factor formula

The scale factor \( k \) of a dilation is the ratio of the length of the image segment to the length of the original segment. For a dilation with center \( Q \), we can use the distances from \( Q \) to the points. Let's find \( QS' \) and \( QS \), or \( QU' \) and \( QU \).

From the diagram, \( QS = 4 \), \( QS' = 4 + 4 = 8 \)? Wait, no, wait. Wait, looking at the diagram: the length from \( Q \) to \( S \) is 4, and from \( Q \) to \( S' \) is \( 4 + 4 = 8 \)? Wait, no, maybe I misread. Wait, the segment from \( S \) to \( S' \) is 4, and from \( Q \) to \( S \) is 4? Wait, no, let's check the other segment. \( QU = 5 \), \( QU' = 5 + 5 = 10 \)? Wait, no, the length of \( SU \) and \( S'U' \)? Wait, no, dilation scale factor is \( \frac{\text{length of image}}{\text{length of pre - image}} \). Alternatively, since \( Q \) is the center, the ratio of \( QS' \) to \( QS \) (or \( QU' \) to \( QU \)) is the scale factor.

Looking at the diagram: \( QS = 4 \), \( QS' = 4 + 4 = 8 \)? Wait, no, the distance from \( Q \) to \( S \) is 4, and from \( Q \) to \( S' \) is \( 4+4 = 8 \)? Wait, no, maybe the length of \( QS \) is 4, and \( QS' \) is \( 4\times2 = 8 \)? Wait, or maybe the length of \( S'U' \) and \( SU \). Wait, \( SU \): let's see, the distance from \( S \) to \( U \): but maybe using the distances from \( Q \). Let's take \( QU = 5 \), \( QU' = 5 + 5 = 10 \). Then the scale factor \( k=\frac{QU'}{QU}=\frac{10}{5} = 2 \). Alternatively, \( QS = 4 \), \( QS' = 4+4 = 8 \), so \( k=\frac{QS'}{QS}=\frac{8}{4}=2 \).

Step2: Calculate the scale factor

We can use the formula for the scale factor of dilation: \( \text{Scale Factor}=\frac{\text{Length of image segment from center}}{\text{Length of pre - image segment from center}} \).

Taking the segments from \( Q \): \( QS = 4 \), \( QS'=4 + 4=8 \). Then the scale factor \( k=\frac{QS'}{QS}=\frac{8}{4} = 2 \).

Or taking \( QU = 5 \), \( QU'=5 + 5 = 10 \), then \( k=\frac{QU'}{QU}=\frac{10}{5}=2 \).

Answer:

\( 2 \)