Question

On a coordinate plane, a line segment contains points (negative 4, 4) and (4, 2). Another line segment contains points (negative 2, 2) and (2, 1).
The ordered pair (–4, 4) is on the pre-image. A scale factor was applied to produce the image on the coordinate plane.

What is the scale factor?
1/4
1/2
2
4

Answer 1

To find the scale factor between the pre-image and the image of the line segments, we can calculate the lengths of the segments before and after scaling.

**Step 1: Calculate the length of the first line segment (pre-image)**

The points are (-4, 4) and (4, 2). The length \(L_1\) is calculated using the distance formula:

\[
L_1 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Applying the points (-4, 4) and (4, 2):

\[
L_1 = \sqrt{(4 - (-4))^2 + (2 - 4)^2} = \sqrt{(4 + 4)^2 + (2 - 4)^2}
\]
\[
= \sqrt{8^2 + (-2)^2} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17}
\]

**Step 2: Calculate the length of the second line segment (image)**

The points are (-2, 2) and (2, 1). The length \(L_2\) is calculated in the same way:

\[
L_2 = \sqrt{(2 - (-2))^2 + (1 - 2)^2}
\]
\[
= \sqrt{(2 + 2)^2 + (1 - 2)^2} = \sqrt{4^2 + (-1)^2}
\]
\[
= \sqrt{16 + 1} = \sqrt{17}
\]

**Step 3: Calculate the scale factor**

The scale factor \(k\) can be found by taking the ratio of the lengths:

\[
k = \frac{L_2}{L_1} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2}
\]

Thus, the scale factor is

\(\boxed{\frac{1}{2}}\).