1. **State the problem:** Dilate figure PQRS by a scale factor of $\frac{1}{2}$ with the center of dilation at point P. 2. **Understand dilation:** Dilation with center $P$ and scale factor $k=\frac{1}{2}$ means each point $X$ in the figure moves along the ray $PX$ to a new point $X'$ such that: $$PX' = k \times PX = \frac{1}{2} PX$$ 3. **Given distances:** - $Q'P = 4$ units (already given) - $S'P = 3$ units (already given) - $R'S' = 2$ units (given) 4. **Find original distances:** Since $Q'P = \frac{1}{2} QP$, original $QP = 2 \times 4 = 8$ units. Similarly, $SP = 2 \times 3 = 6$ units. 5. **Locate points:** - $P$ is at origin $(0,0)$. - $Q$ is 8 units up from $P$ along vertical axis: $Q = (0,8)$. - $S$ is 6 units right from $P$ along horizontal axis: $S = (6,0)$. - $R$ is directly above $S$ and horizontally right of $Q$, forming a right angle at $R$. 6. **Find coordinates of $R$:** Since $R$ is above $S$ and horizontally right of $Q$, $R$ has coordinates $(6,8)$. 7. **Dilate points $R$, $S$, and $T$ (assuming $T$ is a typo for $Q$ or $R$):** - $R'$ lies on ray $PR$ with $PR' = \frac{1}{2} PR$. Calculate $PR = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ units. So, $PR' = 5$ units. Coordinates of $R'$: $$R' = P + \frac{1}{2} (R - P) = (0,0) + \frac{1}{2} (6,8) = (3,4)$$ - $S'$ lies on ray $PS$ with $PS' = \frac{1}{2} PS = 3$ units. Coordinates of $S'$: $$S' = (0,0) + \frac{1}{2} (6,0) = (3,0)$$ - $Q'$ lies on ray $PQ$ with $PQ' = 4$ units (given). Coordinates of $Q'$: $$Q' = (0,0) + \frac{1}{2} (0,8) = (0,4)$$ 8. **Summary of dilated points:** - $P' = P = (0,0)$ (center of dilation) - $Q' = (0,4)$ - $S' = (3,0)$ - $R' = (3,4)$ 9. **Check segment $R'S'$ length:** $$R'S' = \sqrt{(3-3)^2 + (4-0)^2} = \sqrt{0 + 16} = 4$$ Given $R'S' = 2$ units conflicts with calculation, so likely a typo or misinterpretation. **Final answer:** Coordinates of dilated points are $Q' = (0,4)$, $S' = (3,0)$, and $R' = (3,4)$.