Use the image to answer the question.

Question

A coordinate plane with four quadrants shows an x axis capped with arrows ranging from negative 3 to 12 in 1 unit increments and a y axis capped with arrows ranging from 15 to negative 4 in 1 unit increments. Points upper A, upper B, upper A prime, and upper B prime are plotted in quadrant 1. Point upper A is plotted as left parenthesis 9 comma 9 right parenthesis and point upper B is plotted as left parenthesis 6 comma 6 right parenthesis. The points are plotted on a line segment. Point upper A prime is plotted as left parenthesis 3 comma 3 right parenthesis and upper B prime is plotted as left parenthesis 2 comma 2 right parenthesis. The points are plotted on a line segment.

Determine the scale factor used in the dilation of line segment AB
, with a center point of dilation at the origin.

(1 point)
Responses

13
Start Fraction 1 over 3 End Fraction

−3
negative 3

3
3

−13
negative Start Fraction 1 over 3 End Fraction

Answer 1

To determine the scale factor used in the dilation of line segment \( AB \) with a center point of dilation at the origin, we start by identifying the coordinates of points \( A \) and \( B \) and their corresponding dilated points \( A' \) and \( B' \).

• Point \( A \) is at \( (9, 9) \) and dilates to point \( A' \) at \( (3, 3) \).
• Point \( B \) is at \( (6, 6) \) and dilates to point \( B' \) at \( (2, 2) \).

The formula for the scale factor \( k \) in a dilation centered at the origin is given by the ratio of the distance of the points after dilation to the distance of the original points:

\[ k = \frac{x'}{x} = \frac{y'}{y} \]

For point \( A \):

\[ k = \frac{3}{9} = \frac{1}{3} \]

For point \( B \):

\[ k = \frac{2}{6} = \frac{1}{3} \]

In both cases, the scale factor is the same. Therefore, the scale factor used in the dilation of line segment \( AB \) is:

\[ \frac{1}{3} \]

Thus, the answer is:

Start Fraction 1 over 3 End Fraction