Question

Determine the scale factor used in the dilation of line AB
, with a center point of dilation at the origin.
1 (-3, 4) (2, -4) 2 (-6, 8) (4, -8)

Answer 1

To determine the scale factor used in the dilation of line segment \( AB \) with a center of dilation at the origin, we can consider the coordinates of the points before and after the dilation.

Let's define the points:
- Point \( A \) (before dilation): \( A_1 (-3, 4) \)
- Point \( B \) (before dilation): \( B_1 (2, -4) \)
- Point \( A' \) (after dilation): \( A_2 (-6, 8) \)
- Point \( B' \) (after dilation): \( B_2 (4, -8) \)

The scale factor \( k \) can be calculated by comparing the coordinates of the original points and the dilated points. The formula for dilation about the origin is:

\[
(x', y') = (k \cdot x, k \cdot y)
\]

We can find \( k \) for each point:

1. For point \( A \):
- Original point \( A_1 (-3, 4) \)
- Dilated point \( A_2 (-6, 8) \)

Applying the dilation formula:
\[
-6 = k \cdot (-3) \implies k = \frac{-6}{-3} = 2
\]
\[
8 = k \cdot 4 \implies k = \frac{8}{4} = 2
\]

Thus, \( k = 2 \) for point \( A \).

2. For point \( B \):
- Original point \( B_1 (2, -4) \)
- Dilated point \( B_2 (4, -8) \)

Applying the dilation formula:
\[
4 = k \cdot 2 \implies k = \frac{4}{2} = 2
\]
\[
-8 = k \cdot (-4) \implies k = \frac{-8}{-4} = 2
\]

Thus, \( k = 2 \) for point \( B \).

Since the scale factor \( k \) is consistent for both points before and after dilation, we conclude that the scale factor for the dilation of line segment \( AB \) is

\[
\boxed{2}.
\]