1. **State the problem:** We have triangle PQR with vertices P(3,4), Q(5,3), and R(4,1). We want to find the image of this triangle after an enlargement centered at the origin with scale factors (i) $k=2$ and (ii) $k=-\frac{1}{2}$. 2. **Formula for enlargement:** If a point has coordinates $(x,y)$, its image after enlargement with center at the origin and scale factor $k$ is given by: $$ (x', y') = (kx, ky) $$ 3. **Apply the formula for (i) $k=2$:** - For $P(3,4)$: $P' = (2 \times 3, 2 \times 4) = (6, 8)$ - For $Q(5,3)$: $Q' = (2 \times 5, 2 \times 3) = (10, 6)$ - For $R(4,1)$: $R' = (2 \times 4, 2 \times 1) = (8, 2)$ 4. **Apply the formula for (ii) $k=-\frac{1}{2}$:** - For $P(3,4)$: $P' = \left(-\frac{1}{2} \times 3, -\frac{1}{2} \times 4\right) = \left(-\frac{3}{2}, -2\right)$ - For $Q(5,3)$: $Q' = \left(-\frac{1}{2} \times 5, -\frac{1}{2} \times 3\right) = \left(-\frac{5}{2}, -\frac{3}{2}\right)$ - For $R(4,1)$: $R' = \left(-\frac{1}{2} \times 4, -\frac{1}{2} \times 1\right) = (-2, -\frac{1}{2})$ 5. **Summary:** - For $k=2$, the image vertices are $P'(6,8)$, $Q'(10,6)$, $R'(8,2)$. - For $k=-\frac{1}{2}$, the image vertices are $P'\left(-\frac{3}{2}, -2\right)$, $Q'\left(-\frac{5}{2}, -\frac{3}{2}\right)$, $R'\left(-2, -\frac{1}{2}\right)$. This completes the enlargement transformations for both scale factors.