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
, with a center point of dilation at the origin.

(1 point)
Responses

3
3

−3
negative 3

−13
negative Start Fraction 1 over 3 End Fraction

13
Start Fraction 1 over 3 End Fraction

Answer 1

To determine the scale factor used in the dilation of line segment \( AB \) with respect to the origin, we start by finding the lengths of the segments before and after the dilation.

1. **Find the coordinates of points A and B:**
- Point \( A \) is at \( (9, 9) \).
- Point \( B \) is at \( (6, 6) \).

2. **Calculate the distance between points A and B:**
The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting \( A(9, 9) \) and \( B(6, 6) \):
\[
d_{AB} = \sqrt{(6 - 9)^2 + (6 - 9)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]

3. **Now find the coordinates of points \( A' \) and \( B' \):**
- Point \( A' \) is at \( (3, 3) \).
- Point \( B' \) is at \( (2, 2) \).

4. **Calculate the distance between points \( A' \) and \( B' \):**
Again using the distance formula:
\[
d_{A'B'} = \sqrt{(2 - 3)^2 + (2 - 3)^2} = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]

5. **Determine the scale factor (k) for the dilation:**
The scale factor is calculated by the ratio of the distance of \( AB \) to the distance of \( A'B' \):
\[
k = \frac{d_{A'B'}}{d_{AB}} = \frac{\sqrt{2}}{3\sqrt{2}} = \frac{1}{3}
\]

Thus, the scale factor used in the dilation of line segment \( AB \) from \( A \) and \( B \) to \( A' \) and \( B' \) with respect to the origin is:
\[
\frac{1}{3}
\]

The correct response is:
\[
\frac{1}{3}
\]