1. **Problem Statement:** We have a rectangle with top-left corner at $(20, 20)$, width $80$ units (x-direction), and height $60$ units (y-direction downward). 2. **Determine the coordinates of other corners:** - Top-left corner: $(20, 20)$ (given) - Top-right corner: move $80$ units right in x-direction: $(20 + 80, 20) = (100, 20)$ - Bottom-left corner: move $60$ units down in y-direction: $(20, 20 - 60) = (20, -40)$ - Bottom-right corner: move $80$ units right and $60$ units down: $(100, -40)$ 3. **Diagonal length using vector methods:** - Vector from top-left to bottom-right: $\vec{d} = (100 - 20, -40 - 20) = (80, -60)$ - Length of diagonal $d = \sqrt{80^2 + (-60)^2} = \sqrt{6400 + 3600} = \sqrt{10000} = 100$ 4. **Area calculation using coordinate geometry and vector methods:** - Area of rectangle = width $\times$ height = $80 \times 60 = 4800$ - Using vectors, area = magnitude of cross product of adjacent sides: - Vector $\vec{w} = (80, 0)$ (width vector) - Vector $\vec{h} = (0, -60)$ (height vector) - Cross product magnitude $= |80 \times (-60) - 0 \times 0| = 4800$ **Final answers:** - Other corners: $(100, 20)$, $(20, -40)$, $(100, -40)$ - Diagonal length: $100$ - Area: $4800$