Question: 4. Noor needs to dilate the line shown by a scale factor of $\frac{1}{2}$ about the origin. $A = (1, 4)$ $B = (8, -1)$ Preimage A line segment in the center-left to lower-right of the graph is labeled Preimage, with endpoints $A (1, 4)$ near the top-left of the graph and $B (8, -1)$ near the lower-right. Will the dilated line lie on top of the preimage? No, the scale factor is less than 1. Yes, the center of dilation is the origin. No, the line segment does not pass through the center of dilation. Yes, the line segment does not touch the origin.

1. **State the problem:** Noor needs to dilate the line segment with endpoints $A(1,4)$ and $B(8,-1)$ by a scale factor of $\frac{1}{2}$ about the origin. 2. **Recall the dilation formula:** Dilation about the origin with scale factor $k$ transforms a point $(x,y)$ to $(kx, ky)$. 3. **Apply the dilation to each endpoint:** - For $A(1,4)$, the dilated point $A' = \left(\frac{1}{2} \times 1, \frac{1}{2} \times 4\right) = \left(\frac{1}{2}, 2\right)$. - For $B(8,-1)$, the dilated point $B' = \left(\frac{1}{2} \times 8, \frac{1}{2} \times (-1)\right) = (4, -\frac{1}{2})$. 4. **Check if the dilated line lies on top of the preimage:** - The scale factor $\frac{1}{2}$ is less than 1, so the dilated segment is smaller and lies closer to the origin. - The original segment does not pass through the origin, so the dilated segment will not coincide with the original. 5. **Conclusion:** - The dilated line segment $A'B'$ has endpoints $\left(\frac{1}{2}, 2\right)$ and $(4, -\frac{1}{2})$. - It does not lie on top of the preimage. - The center of dilation is the origin. - The original segment does not pass through the origin. **Final answer:** The dilated line segment does not lie on top of the preimage because the scale factor is less than 1 and the segment does not pass through the origin.