1. **State the problem:** We are given a rectangle with midpoints M and N on two sides and need to find the coordinates of the center C of the rectangle. 2. **Recall properties:** The center C of a rectangle is the midpoint of the diagonal connecting opposite corners. 3. **Given points:** - M is the midpoint of the right side at (20, 11) - N is the midpoint of the bottom side at (13, 7) 4. **Find the coordinates of the rectangle's corners using midpoints:** - Since M is midpoint of the right side, the right side's vertical coordinate range is symmetric about M. - Since N is midpoint of the bottom side, the bottom side's horizontal coordinate range is symmetric about N. 5. **Let the rectangle's corners be:** - Bottom-right corner: $B = (x_B, y_B)$ - Top-right corner: $T = (x_T, y_T)$ - Bottom-left corner: $L = (x_L, y_L)$ - Top-left corner: $U = (x_U, y_U)$ 6. **From M (midpoint of right side):** - $M = \left(\frac{x_B + x_T}{2}, \frac{y_B + y_T}{2}\right) = (20, 11)$ 7. **From N (midpoint of bottom side):** - $N = \left(\frac{x_B + x_L}{2}, \frac{y_B + y_L}{2}\right) = (13, 7)$ 8. **Since rectangle sides are parallel to axes:** - Right side vertical line: $x_B = x_T = 20 \times 2 - x_T$ (but we need to find $x_L$ and $x_U$) - Bottom side horizontal line: $y_B = y_L = 7 \times 2 - y_L$ 9. **Use midpoint formulas to find corners:** - From M: $20 = \frac{x_B + x_T}{2}$ and $11 = \frac{y_B + y_T}{2}$ - From N: $13 = \frac{x_B + x_L}{2}$ and $7 = \frac{y_B + y_L}{2}$ 10. **Since right side vertical line means $x_B = x_T = 20$ (because M is midpoint of right side), so:** - $x_B = x_T = 20$ 11. **From N's x-coordinate:** - $13 = \frac{20 + x_L}{2} \Rightarrow 26 = 20 + x_L \Rightarrow x_L = 6$ 12. **Since bottom side horizontal line means $y_B = y_L = 7$ (because N is midpoint of bottom side), so:** - $y_B = y_L = 7$ 13. **From M's y-coordinate:** - $11 = \frac{7 + y_T}{2} \Rightarrow 22 = 7 + y_T \Rightarrow y_T = 15$ 14. **Now we have corners:** - Bottom-right $B = (20, 7)$ - Top-right $T = (20, 15)$ - Bottom-left $L = (6, 7)$ 15. **Find top-left corner $U$ coordinates:** - Since rectangle, $U$ shares x with $L$ and y with $T$: - $U = (6, 15)$ 16. **Find center C as midpoint of diagonal $B$ and $U$:** $$ C = \left(\frac{20 + 6}{2}, \frac{7 + 15}{2}\right) = (13, 11) $$ **Final answer:** The coordinates of the center $C$ are **(13, 11)**.