1. The problem asks for the measure of angle A' B' C' after a dilation of right angle ABC with center P and scale factor $\frac{1}{2}$. 2. Important rule: Dilation changes the size of a figure but preserves angle measures. 3. Since angle ABC is a right angle, its measure is $90^\circ$. 4. After dilation, angle A' B' C' will have the same measure as angle ABC because dilation preserves angles. 5. Therefore, the measure of angle A' B' C' is $90^\circ$. 6. Next, dilate point C using center D and scale factor $\frac{3}{4}$. 7. The dilation formula for a point $X$ with center $D$ and scale factor $k$ is: $$X' = D + k(X - D)$$ 8. This means the vector from $D$ to $X$ is scaled by $k$. 9. For point C, the dilated point $C'$ is: $$C' = D + \frac{3}{4}(C - D)$$ 10. For dilating segment AB using center D and scale factor $\frac{1}{2}$, each endpoint of segment AB is dilated similarly: $$A' = D + \frac{1}{2}(A - D)$$ $$B' = D + \frac{1}{2}(B - D)$$ 11. The dilated segment A'B' is formed by points A' and B'. 12. The length of segment A'B' is half the length of AB because scale factor is $\frac{1}{2}$. Final answers: - Measure of angle A' B' C' is $90^\circ$. - Dilated point C' is at $D + \frac{3}{4}(C - D)$. - Dilated segment A'B' has endpoints $A' = D + \frac{1}{2}(A - D)$ and $B' = D + \frac{1}{2}(B - D)$ with length half of AB.