1. **Problem Statement:** We are given a pentagon ABCDE with right angles at vertices B, D, and E. The lengths AE = AB = 5, BC = 12, and DC = DE. We need to find the area of pentagon ABCDE. 2. **Understanding the shape:** The pentagon has right angles at B, D, and E, which suggests some sides are perpendicular. AE and AB are equal, both 5 units. BC is 12 units. Since DC = DE, triangle DCE is isosceles with right angle at D or E. 3. **Approach:** We can divide the pentagon into simpler shapes (triangles and rectangles) to find the total area. 4. **Step 1: Coordinates assignment for clarity:** - Place point A at origin (0,0). - Since AB = 5 and right angle at B, let B be at (5,0). - Since AE = 5 and right angle at E, let E be at (0,5). 5. **Step 2: Locate point C:** - BC = 12 and right angle at B means BC is vertical or horizontal. - Since B is at (5,0), and right angle at B, C must be at (5,12) (vertical segment). 6. **Step 3: Locate points D and E:** - E is at (0,5). - DC = DE and right angle at D and E. - Since D is between C(5,12) and E(0,5), and DC = DE, D lies on the line segment CE such that DC = DE. 7. **Step 4: Find coordinates of D:** - Vector CE = E - C = (0-5,5-12) = (-5,-7). - Let D divide CE into two equal segments since DC = DE, so D is midpoint of CE. - Midpoint D = ((5+0)/2, (12+5)/2) = (2.5, 8.5). 8. **Step 5: Calculate area of pentagon ABCDE:** - Use the shoelace formula with points in order A(0,0), B(5,0), C(5,12), D(2.5,8.5), E(0,5). Shoelace formula: $$\text{Area} = \frac{1}{2} |x_1y_2 + x_2y_3 + x_3y_4 + x_4y_5 + x_5y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_5 + y_5x_1)|$$ Calculate: $$x_1y_2 = 0 \times 0 = 0$$ $$x_2y_3 = 5 \times 12 = 60$$ $$x_3y_4 = 5 \times 8.5 = 42.5$$ $$x_4y_5 = 2.5 \times 5 = 12.5$$ $$x_5y_1 = 0 \times 0 = 0$$ Sum1 = 0 + 60 + 42.5 + 12.5 + 0 = 115$$ $$y_1x_2 = 0 \times 5 = 0$$ $$y_2x_3 = 0 \times 5 = 0$$ $$y_3x_4 = 12 \times 2.5 = 30$$ $$y_4x_5 = 8.5 \times 0 = 0$$ $$y_5x_1 = 5 \times 0 = 0$$ Sum2 = 0 + 0 + 30 + 0 + 0 = 30$$ Area: $$\frac{1}{2} |115 - 30| = \frac{1}{2} \times 85 = 42.5$$ 9. **Final answer:** The area of pentagon ABCDE is **42.5**.