Question

A(9,9) B(6,6)
A(3,3) B(2,2)
Determine the scale factor used in the dilation of line segment AB, with a center point of a dilation at the origin.

(1 point)

-3
3
-⅓
⅓

Answer 1

To determine the scale factor used in the dilation of line segment AB with respect to the origin, we need to examine the coordinates of points A and B and find their dilated positions.

Let's first determine the length of segment AB with the original coordinates provided.

1. For the first pair of points: A(9, 9) and B(6, 6).

- The distance \( AB \) is calculated as:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(6 - 9)^2 + (6 - 9)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]

2. For the second pair of points: A(3, 3) and B(2, 2).

- The distance \( AB \) is calculated as:
\[
d = \sqrt{(2 - 3)^2 + (2 - 3)^2} = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]

Next, we can compare the distances of the two segments to determine the scale factor.

The scale factor \( k \) is given by the ratio of the lengths of the segments:
\[
k = \frac{\text{length of new segment (AB)}}{\text{length of original segment (AB)}}
\]
\[
k = \frac{\sqrt{2}}{3\sqrt{2}} = \frac{1}{3}
\]

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

So the correct answer is:

\( \frac{1}{3} \) (or ⅓)