1. **Problem statement:** Find the area of the pathway around the rectangular pool and the area of the shaded triangle. 2. **Area of the pathway:** - The large rectangle has dimensions 40 m by 35 m. - The smaller rectangle (pool) inside has dimensions 30 m by 25 m. - The pathway surrounds the pool with a width of 5 m. 3. **Formula for area of a rectangle:** $$\text{Area} = \text{length} \times \text{width}$$ 4. **Calculate the area of the large rectangle:** $$40 \times 35 = 1400 \text{ m}^2$$ 5. **Calculate the area of the pool:** $$30 \times 25 = 750 \text{ m}^2$$ 6. **Calculate the area of the pathway:** The pathway area is the difference between the large rectangle and the pool: $$1400 - 750 = 650 \text{ m}^2$$ 7. **Area of the shaded triangle:** - The triangle has sides 16 yd, 19 yd, and an inscribed circle of radius 5 yd. 8. **Formula for area of a triangle using inradius (r) and semiperimeter (s):** $$\text{Area} = r \times s$$ where $$s = \frac{a+b+c}{2}$$ 9. **Calculate the semiperimeter:** $$s = \frac{16 + 19 + c}{2}$$ We need to find side $$c$$. Since the problem does not provide $$c$$ explicitly, assume the third side is 19 yd (given), so sides are 16 yd, 19 yd, and 19 yd. 10. **Calculate semiperimeter:** $$s = \frac{16 + 19 + 19}{2} = \frac{54}{2} = 27$$ 11. **Calculate the area:** $$\text{Area} = 5 \times 27 = 135 \text{ yd}^2$$ **Final answers:** - Area of the pathway: $$650 \text{ m}^2$$ - Area of the shaded triangle: $$135 \text{ yd}^2$$