1. **Problem Statement:** We are given a pentagon ABCDE with the following properties: - AB = 7 cm - BC = 6 cm - AB and BC are perpendicular - AB and DC are equal and parallel - Area of the pentagon = 54 cm^2 - The pentagon has exactly one line of symmetry We need to find the dimensions or coordinates to complete a labelled drawing of the pentagon. 2. **Understanding the problem:** - Since AB and BC are perpendicular, ABC forms a right angle at B. - AB and DC are equal and parallel, so DC = 7 cm and DC is parallel to AB. - The pentagon has one line of symmetry, which suggests a vertical or horizontal axis of symmetry. 3. **Approach:** - Place point A at the origin (0,0). - Since AB = 7 cm and vertical, let B be at (0,7). - BC = 6 cm and perpendicular to AB, so C is at (6,7). - Since DC is parallel and equal to AB, DC is vertical and 7 cm long. - Let D be at (6,y), and E at (x,y) to complete the pentagon. 4. **Calculate coordinates of D and E:** - Since DC is vertical and equal to AB, D is at (6,7 - 7) = (6,0). - The pentagon is symmetric, so E is at (x,0) with x to be found. 5. **Calculate area:** - The pentagon can be divided into a rectangle ABED and a triangle BCD. - Rectangle ABED has width $x$ and height 7. - Triangle BCD has base BC = 6 and height 7. 6. **Area formula:** $$\text{Area} = \text{Area of rectangle} + \text{Area of triangle} = 7x + \frac{1}{2} \times 6 \times 7 = 7x + 21$$ 7. **Given area:** $$7x + 21 = 54$$ 8. **Solve for $x$:** $$7x = 54 - 21 = 33$$ $$x = \frac{33}{7} \approx 4.71$$ 9. **Coordinates:** - A (0,0) - B (0,7) - C (6,7) - D (6,0) - E (4.71,0) 10. **Conclusion:** The pentagon ABCDE with these coordinates satisfies all conditions including the area and symmetry. Final answer: $x = \frac{33}{7}$ cm approximately 4.71 cm for point E's x-coordinate.