1. **State the problem:** We have a line segment \(\overline{PQ}\) with endpoints \(P(0,4)\) and \(Q(-6,4)\). It is dilated from the origin with a scale factor of \(\frac{3}{2}\). We need to find the coordinates of the image points \(P'\) and \(Q'\), and the length of the image segment \(\overline{P'Q'}\). 2. **Formula for dilation:** If a point \((x,y)\) is dilated from the origin by scale factor \(k\), the image point is \((kx, ky)\). 3. **Apply dilation to points:** - For \(P(0,4)\), \[ P' = \left( \frac{3}{2} \times 0, \frac{3}{2} \times 4 \right) = (0, 6) \] - For \(Q(-6,4)\), \[ Q' = \left( \frac{3}{2} \times (-6), \frac{3}{2} \times 4 \right) = (-9, 6) \] 4. **Calculate original length \(\overline{PQ}\):** Since \(P\) and \(Q\) have the same \(y\)-coordinate, length is difference in \(x\): \[ \text{length} = |0 - (-6)| = 6 \] 5. **Calculate image length \(\overline{P'Q'}\):** Length scales by the scale factor \(k = \frac{3}{2}\), so \[ \text{length of } \overline{P'Q'} = 6 \times \frac{3}{2} = 9 \] **Final answers:** - \(P' = (0, 6)\) - \(Q' = (-9, 6)\) - Length of \(\overline{P'Q'} = 9\)