Question

Use the image to answer the 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
A
B
, with a center point of dilation at the origin.

(1 point)
Responses

−13
−
1
3
negative Start Fraction 1 over 3 End Fraction

3
3

−3
−
3
negative 3

13

Answer 1

To determine the scale factor used in the dilation of line segment AB with center point of dilation at the origin, we need to compare the coordinates of the points before and after the dilation.

1. Original points (before dilation):
- Point A: \( A(9, 9) \)
- Point B: \( B(6, 6) \)

2. New points (after dilation):
- Point A': \( A'(3, 3) \)
- Point B': \( B'(2, 2) \)

The scale factor \( k \) can be found by comparing the distances from the origin (0, 0) to each set of points.

For point A:
\[
k = \frac{\text{Distance from origin to } A'}{\text{Distance from origin to } A} = \frac{\sqrt{3^2 + 3^2}}{\sqrt{9^2 + 9^2}} = \frac{\sqrt{9 + 9}}{\sqrt{81 + 81}} = \frac{\sqrt{18}}{\sqrt{162}} = \frac{3\sqrt{2}}{9\sqrt{2}} = \frac{3}{9} = \frac{1}{3}
\]

For point B:
\[
k = \frac{\text{Distance from origin to } B'}{\text{Distance from origin to } B} = \frac{\sqrt{2^2 + 2^2}}{\sqrt{6^2 + 6^2}} = \frac{\sqrt{4 + 4}}{\sqrt{36 + 36}} = \frac{\sqrt{8}}{\sqrt{72}} = \frac{2\sqrt{2}}{6\sqrt{2}} = \frac{2}{6} = \frac{1}{3}
\]

Since both points A and B give the same scale factor, the scale factor for the dilation is \( \frac{1}{3} \).

Thus, the answer is:
**Start Fraction 1 over 3 End Fraction**