Question

line segment ab is dilated to create line segment ab using point q as the center of dilation. what is the scale factor? the image shows a diagram with points q, a, a, b, b and lengths 1.25 from q to a, 1.25 from a to a, and multiple choice options 1.25, 2.5, 1, 2

Explanation:

Step1: Recall dilation scale factor formula

The scale factor \( k \) of a dilation is given by \( k=\frac{\text{Length of image segment}}{\text{Length of original segment}} \), or using distances from the center of dilation: \( k = \frac{QA'}{QA} \).

Step2: Identify \( QA \) and \( QA' \)

From the diagram, \( QA = 1.25 \) and \( QA'=QA + AA'=1.25 + 1.25 = 2.5 \)? Wait, no, wait. Wait, actually, looking at the diagram, \( QA \) is 1.25? Wait, no, maybe I misread. Wait, the segments: \( QA \) is 1.25? Wait, no, the length from \( Q \) to \( A \) is 1.25, and from \( Q \) to \( A' \) is \( QA + AA' = 1.25+1.25 = 2.5 \)? Wait, no, wait the scale factor is \( \frac{QA'}{QA} \). Wait, let's check again. Wait, the original segment is \( AB \), and the image is \( A'B' \). But using the center of dilation \( Q \), the scale factor is the ratio of the distance from \( Q \) to a point on the image (\( A' \)) to the distance from \( Q \) to the corresponding point on the original (\( A \)). So \( QA = 1.25 \)? Wait, no, the diagram shows \( QA \) as 1.25? Wait, no, the labels: \( QA \) is 1.25? Wait, no, the length from \( Q \) to \( A \) is 1.25, and from \( Q \) to \( A' \) is \( 1.25 + 1.25 = 2.5 \)? Wait, no, maybe \( QA \) is 1.25, and \( QA' \) is \( 1.25\times2 = 2.5 \)? Wait, no, let's calculate the scale factor. The scale factor \( k=\frac{QA'}{QA} \). Let's find \( QA \) and \( QA' \). From the diagram, \( QA = 1.25 \)? Wait, no, maybe \( QA \) is 1.25, and \( QA' \) is \( 1.25 + 1.25 = 2.5 \)? Wait, no, that would make \( k = \frac{2.5}{1.25}=2 \)? But that's not matching. Wait, maybe I misread the diagram. Wait, the problem says line segment \( AB \) is dilated to \( A'B' \) with center \( Q \). Let's look at the lengths. Wait, the distance from \( Q \) to \( A \) is 1.25, and from \( Q \) to \( A' \) is \( 1.25 + 1.25 = 2.5 \)? Wait, no, maybe the length of \( QA \) is 1.25, and \( QA' \) is \( 1.25\times2 = 2.5 \), so scale factor \( k=\frac{QA'}{QA}=\frac{2.5}{1.25}=2 \)? But the options include 2. Wait, but let's check again. Wait, maybe the original \( QA \) is 1.25, and \( QA' \) is \( 1.25 + 1.25 = 2.5 \), so \( k = 2.5 / 1.25 = 2 \). Wait, but let's confirm. Alternatively, maybe the length of \( AB \) and \( A'B' \). But since it's a dilation, the ratio of \( QA' \) to \( QA \) is the scale factor. So \( QA = 1.25 \), \( QA' = 1.25 + 1.25 = 2.5 \), so \( k = 2.5 / 1.25 = 2 \). Wait, but the options have 2 as one of them. Wait, but let's check the diagram again. Wait, the labels: \( QA \) is 1.25, \( AA' \) is 1.25, so \( QA' = QA + AA' = 1.25 + 1.25 = 2.5 \). Then scale factor \( k = \frac{QA'}{QA} = \frac{2.5}{1.25} = 2 \). Wait, but maybe I made a mistake. Wait, another way: the scale factor is \( \frac{\text{length of } A'B'}{\text{length of } AB} \). But since \( AB \) and \( A'B' \) are parallel (because dilation preserves parallelism), triangles \( QAB \) and \( QA'B' \) are similar, so the ratio of corresponding sides is the scale factor. So \( \frac{QA'}{QA}=\frac{QB'}{QB}=k \). From the diagram, \( QA = 1.25 \), \( QA' = 1.25 + 1.25 = 2.5 \), so \( k = 2.5 / 1.25 = 2 \). So the scale factor is 2.

Answer:

2 (the option with value 2)