y COURSE TOOLS 10- 9- 8- 7- 6- X < < 5- 4- 3- 2- 1- 10+ -10-9-8-7-6-5-4-3-2-10 B XY is dilated with a scale factor of 2 and a center of dilation at the origin. Find points X' and Y' to prove that a dilation of 2 has a length that is twice the length of XY (1 point) X' is at (-4, 1) and Y' is at (-12, 5) OX' is at (-6, 8) and Y' is at (-14, 4) X' is at (-1.5,2) and Y' is at (-3.5.1) X' is at (6, -8) and Y' is at (14, -4)

To find the points \( X' \) and \( Y' \) after a dilation of scale factor 2 with the center of dilation at the origin, we need to apply the dilation formula:

\[
X' = k \cdot X
\]
\[
Y' = k \cdot Y
\]

where \( k \) is the scale factor, and \( X \) and \( Y \) are the original points.

However, we first need to know the original points \( X \) and \( Y \). Unfortunately, they are not provided, but we can use the transformations to evaluate the given options.

1. **Option 1:**
- \( X' = (-4, 1) \)
- \( Y' = (-12, 5) \)

The original points would be \( X = \left( \frac{-4}{2}, \frac{1}{2} \right) = (-2, 0.5) \) and \( Y = \left( \frac{-12}{2}, \frac{5}{2} \right) = (-6, 2.5) \).

2. **Option 2:**
- \( X' = (-6, 8) \)
- \( Y' = (-14, 4) \)

The original points would be \( X = \left( \frac{-6}{2}, \frac{8}{2} \right) = (-3, 4) \) and \( Y = \left( \frac{-14}{2}, \frac{4}{2} \right) = (-7, 2) \).

3. **Option 3:**
- \( X' = (-1.5, 2) \)
- \( Y' = (-3.5, 1) \)

The original points would be \( X = \left( \frac{-1.5}{2}, \frac{2}{2} \right) = (-0.75, 1) \) and \( Y = \left( \frac{-3.5}{2}, \frac{1}{2} \right) = (-1.75, 0.5) \).

4. **Option 4:**
- \( X' = (6, -8) \)
- \( Y' = (14, -4) \)

The original points would be \( X = \left( \frac{6}{2}, \frac{-8}{2} \right) = (3, -4) \) and \( Y = \left( \frac{14}{2}, \frac{-4}{2} \right) = (7, -2) \).

To prove that the length of \( X'Y' \) is twice the length of \( XY \), we can use the distance formula:

\[
\text{Distance between } A(x_1, y_1) \text{ and } B(x_2, y_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### For Option 1:
- Distance \( XY = \sqrt{((-6) - (-2))^2 + ((2.5) - (0.5))^2} = \sqrt{(-4)^2 + (2)^2} = \sqrt{16 + 4} = \sqrt{20} \)
- Distance \( X'Y' = \sqrt{((-12) - (-4))^2 + ((5) - (1))^2} = \sqrt{(-8)^2 + (4)^2} = \sqrt{64 + 16} = \sqrt{80} \)

Since \( \sqrt{80} = 4\sqrt{5} \) and \( \sqrt{20} = 2\sqrt{5} \), we can clearly see that \( \sqrt{80} = 2\sqrt{20} \): so this option works.

The conclusion is that **Option 1** is indeed correct for dilation when \( k = 2 \) at the origin.