1. **Stating the problem:** We need to construct the images of the line segment [AB] under enlargements with center P and scale factors 2, 3, 3/2, and 1/2. 2. **Formula for enlargement:** If $P$ is the center of enlargement and $A$, $B$ are points, the image $A'$ of $A$ is given by: $$A' = P + k(A - P)$$ where $k$ is the scale factor. 3. **Step-by-step construction for each scale factor:** (i) Scale factor $k=2$: $$A' = P + 2(A - P) = P + 2A - 2P = 2A - P$$ $$B' = P + 2(B - P) = 2B - P$$ (ii) Scale factor $k=3$: $$A' = P + 3(A - P) = 3A - 2P$$ $$B' = P + 3(B - P) = 3B - 2P$$ (iii) Scale factor $k=\frac{3}{2}$: $$A' = P + \frac{3}{2}(A - P) = P + \frac{3}{2}A - \frac{3}{2}P = \frac{3}{2}A - \frac{1}{2}P$$ $$B' = P + \frac{3}{2}(B - P) = \frac{3}{2}B - \frac{1}{2}P$$ (iv) Scale factor $k=\frac{1}{2}$: $$A' = P + \frac{1}{2}(A - P) = P + \frac{1}{2}A - \frac{1}{2}P = \frac{1}{2}A + \frac{1}{2}P$$ $$B' = P + \frac{1}{2}(B - P) = \frac{1}{2}B + \frac{1}{2}P$$ 4. **Explanation:** For each scale factor, the image points $A'$ and $B'$ lie on the line through $P$ and the original points $A$ and $B$, respectively. The distance from $P$ to $A'$ (or $B'$) is $k$ times the distance from $P$ to $A$ (or $B$). 5. **Summary:** - For $k=2$, points are twice as far from $P$ as original. - For $k=3$, points are three times as far. - For $k=\frac{3}{2}$, points are 1.5 times as far. - For $k=\frac{1}{2}$, points are half as far. This completes the construction of the images of [AB] under the given enlargements.