Question

Use the image to answer the question.
Determine the scale factor used in the dilation of line AB
𝐴
𝐵
, with a center point of dilation at the origin.

(1 point)
longer -6,8 4,-8 shorter -3,4 2,-4
its not 1/2

Answer 1

To determine the scale factor used in the dilation of line segment AB with the center of dilation at the origin, we need the coordinates of the points A and B before and after the dilation.

Considering you have provided different points, let's break it down:

1. **Identify the original coordinates of points A and B**. It seems like you've provided the points:
- A (original): \((-6, 8)\)
- B (original): \((4, -8)\)

2. **Identify the coordinates of the points after dilation (if provided)**. It seems like you also provided:
- A' (dilated): \((-3, 4)\)
- B' (dilated): \((2, -4)\)

3. **Calculate the scale factor** using the formula for dilation:
\[
\text{Scale Factor} = \frac{\text{Length of segment after dilation}}{\text{Length of segment before dilation}}
\]

4. **Calculate the length of the original line segment AB** and the dilated line segment A'B':
- Length of segment AB:
\[
\text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Using A \((-6, 8)\) and B \((4, -8)\):
\[
AB = \sqrt{(4 - (-6))^2 + (-8 - 8)^2} = \sqrt{(4 + 6)^2 + (-8 - 8)^2} = \sqrt{10^2 + (-16)^2}
\]
\[
= \sqrt{100 + 256} = \sqrt{356}
\]

- Length of segment A'B':
Using A' \((-3, 4)\) and B' \((2, -4)\):
\[
A'B' = \sqrt{(2 - (-3))^2 + (-4 - 4)^2} = \sqrt{(2 + 3)^2 + (-4 - 4)^2} = \sqrt{5^2 + (-8)^2}
\]
\[
= \sqrt{25 + 64} = \sqrt{89}
\]

5. **Calculate the scale factor**:
\[
\text{Scale Factor} = \frac{\sqrt{89}}{\sqrt{356}} = \frac{\sqrt{89}}{2\sqrt{89}} = \frac{1}{2}
\]

If you stated that the scale factor is not \( \frac{1}{2} \), it could be due to a misunderstanding regarding the points before or after dilation. Please double-check the coordinates or share any other details to clarify.