1. **State the problem:** We have rectangle ABCD with points A(3,5), midpoint M(1,5) of AB, and midpoint N(-1,-1) of BC. We need to find coordinates of point D. 2. **Recall properties of a rectangle:** Opposite sides are equal and parallel. Midpoint formula for segment between points $(x_1,y_1)$ and $(x_2,y_2)$ is $$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right).$$ 3. **Find coordinates of B using midpoint M of AB:** $$M = \left(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}\right) = (1,5)$$ Substitute $A=(3,5)$: $$1 = \frac{3 + x_B}{2} \Rightarrow 2 = 3 + x_B \Rightarrow x_B = 2 - 3 = -1$$ $$5 = \frac{5 + y_B}{2} \Rightarrow 10 = 5 + y_B \Rightarrow y_B = 5$$ So, $B=(-1,5)$. 4. **Find coordinates of C using midpoint N of BC:** $$N = \left(\frac{x_B + x_C}{2}, \frac{y_B + y_C}{2}\right) = (-1,-1)$$ Substitute $B=(-1,5)$: $$-1 = \frac{-1 + x_C}{2} \Rightarrow -2 = -1 + x_C \Rightarrow x_C = -1$$ $$-1 = \frac{5 + y_C}{2} \Rightarrow -2 = 5 + y_C \Rightarrow y_C = -7$$ So, $C=(-1,-7)$. 5. **Find coordinates of D:** Since ABCD is a rectangle, $D$ is opposite $B$ relative to $C$ and $A$. Vector $\overrightarrow{AB} = B - A = (-1 - 3, 5 - 5) = (-4, 0)$ Vector $\overrightarrow{BC} = C - B = (-1 + 1, -7 - 5) = (0, -12)$ Vector $\overrightarrow{AD} = \overrightarrow{BC} = (0, -12)$ So, $$D = A + \overrightarrow{AD} = (3,5) + (0,-12) = (3, -7)$$ **Final answer:** The coordinates of point D are $(3, -7)$.