Question

Use the image to answer the question.

A coordinate plane with four quadrants showsthe x-axis ranging from negative 4 to 12 in increments of 1 and the y-axis from 1 to negative 16 in increments of 1. Points upper A, upper B, upper A prime, and upper B prime are plotted on quadrant 4. The plotted points are as follows: Upper A is left parenthesis 1 comma negative 3 right parenthesis and upper B is left parenthesis 2 comma negative 4 right parenthesis. The points are joined by a line. Upper A prime is plotted as left parenthesis 4 comma negative 12 right parenthesis and upper B prime as left parenthesis 8 comma negative 16 right parenthesis. The points are joined by a line.

Determine the scale factor used in the dilation of line segment AB
A
B
, centered at the origin.

(1 point)
Responses

2
2

−4
−
4
negative 4

14
1
4
Start Fraction 1 over 4 End Fraction

4

Answer 1

To determine the scale factor used in the dilation of the line segment \( AB \) centered at the origin, we first identify the coordinates of points \( A \) and \( B \), as well as their corresponding dilated points \( A' \) and \( B' \).

The points are as follows:
- \( A(1, -3) \)
- \( B(2, -4) \)
- \( A'(4, -12) \)
- \( B'(8, -16) \)

Next, we can calculate the scale factor by comparing the coordinates of the dilated points to the original points.

The scale factor \( k \) can be calculated as:

\[
k = \frac{\text{coordinate of } A'}{\text{coordinate of } A}
\]

For point \( A \):

1. For the x-coordinate:
\[
k_x = \frac{4}{1} = 4
\]

2. For the y-coordinate:
\[
k_y = \frac{-12}{-3} = 4
\]

Thus, the scale factor from point \( A \) to \( A' \) is \( 4 \).

Now, let’s check for point \( B \):

1. For the x-coordinate:
\[
k_x = \frac{8}{2} = 4
\]

2. For the y-coordinate:
\[
k_y = \frac{-16}{-4} = 4
\]

The scale factor from point \( B \) to \( B' \) is also \( 4 \).

Since both points \( A \) and \( B \) have the same scale factor of \( 4 \), we conclude that the scale factor used in the dilation of line segment \( AB \) centered at the origin is:

\[
\boxed{4}
\]