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
negative Start Fraction 1 over 3 End Fraction

3
3

−3
negative 3

13

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 can use the coordinates of points \(A\) and \(B\) before dilation and points \(A'\) and \(B'\) after dilation.

1. Identify the coordinates:

   • Point \(A\) has coordinates \(A(9, 9)\).
   • Point \(B\) has coordinates \(B(6, 6)\).
   • Point \(A'\) has coordinates \(A'(3, 3)\).
   • Point \(B'\) has coordinates \(B'(2, 2)\).

2. Calculate the scale factor: The scale factor \(k\) can be calculated using the formula \[ k = \frac{\text{New coordinate}}{\text{Original coordinate}}. \]

   For point \(A\), \[ k_A = \frac{A'(x)}{A(x)} = \frac{3}{9} = \frac{1}{3}. \]

   For point \(B\), \[ k_B = \frac{B'(x)}{B(x)} = \frac{2}{6} = \frac{1}{3}. \]

3. Conclusion: The scale factor is consistent for both points and is \( \frac{1}{3} \).

Thus, the scale factor used in the dilation of line segment \(AB\) is negative \( \frac{1}{3} \).

The correct answer is negative \( \frac{1}{3} \).