Question

A coordinate plane with four quadrants shows an x axis capped with arrows and ranging from negative 11 to 10 in 1 unit increments and a y axis capped with arrows and ranging from 12 to negative 12 in 1 unit increments. Points upper A, upper B, upper A prime, and upper B prime are plotted. A coordinate plane with four quadrants shows an x axis capped with arrows and ranging from negative 11 to 10 in 1 unit increments and a y axis capped with arrows and ranging from 12 to negative 12 in 1 unit increments. Point upper A is plotted as left parenthesis negative 3 comma 4 right parenthesis and point upper B is plotted as left parenthesis 2 comma negative 4 right parenthesis. The points are joined by a line that extends beyond the points and is capped with arrows at both ends. Point upper A prime is plotted as left parenthesis negative 6 comma 8 right parenthesis and point upper B prime is plotted as left parenthesis 4 comma negative 8 right parenthesis. The points are joined by a line that extends beyond the points and is capped with arrows at both ends.
Determine the scale factor used in the dilation of line AB, with a center point of dilation at the origin.

Answer 1

To find the scale factor used in the dilation of line \( AB \) to line \( A'B' \) with the center of dilation at the origin, we can follow these steps:

1. **Identify the coordinates of points A and B**:
Point \( A \) is at \( (-3, 4) \) and point \( B \) is at \( (2, -4) \).

2. **Identify the coordinates of points A' and B'**:
Point \( A' \) is at \( (-6, 8) \) and point \( B' \) is at \( (4, -8) \).

3. **Calculate the distances from the origin to each point**:
- For point \( A \):
\[
\text{Distance to } A = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
- For point \( B \):
\[
\text{Distance to } B = \sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]
- For point \( A' \):
\[
\text{Distance to } A' = \sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
- For point \( B' \):
\[
\text{Distance to } B' = \sqrt{4^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}
\]

4. **Calculate the scale factor**:
The scale factor \( k \) can be found by comparing the distances from the origin to the points before and after dilation. The distances scale as follows:
- For point \( A \) to \( A' \):
\[
k = \frac{\text{Distance to } A'}{\text{Distance to } A} = \frac{10}{5} = 2
\]
- For point \( B \) to \( B' \):
\[
k = \frac{\text{Distance to } B'}{\text{Distance to } B} = \frac{4\sqrt{5}}{2\sqrt{5}} = 2
\]

Since both points give the same scale factor, we confirm that the scale factor used in the dilation from line \( AB \) to line \( A'B' \) is:

\[
\boxed{2}
\]