1. **Problem Statement:** Calculate the perimeter and area of two composite figures. --- ### a) Irregular pentagon-like shape with sides 5 m (top), 6 m (right), 9 m (bottom), 3 m (left), and a slanted upper-left side. 2. **Perimeter Calculation:** - The perimeter is the sum of all side lengths. - We know four sides: 5 m, 6 m, 9 m, 3 m. - We need to find the length of the slanted upper-left side. 3. **Finding the slanted side length:** - Assume the figure is composed of a rectangle and a right triangle on the upper-left. - The vertical side is 3 m, horizontal top side is 5 m. - The bottom side is 9 m, so the difference between 9 m and 5 m is 4 m, which is the horizontal leg of the triangle. - The vertical leg is 3 m. - Use the Pythagorean theorem: $$\text{slanted side} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$ 4. **Sum all sides for perimeter:** $$P = 5 + 6 + 9 + 3 + 5 = 28\text{ m}$$ 5. **Area Calculation:** - Split the figure into a rectangle and a right triangle. - Rectangle area: base 5 m, height 6 m $$A_{rect} = 5 \times 6 = 30\text{ m}^2$$ - Triangle area: base 4 m, height 3 m $$A_{tri} = \frac{1}{2} \times 4 \times 3 = 6\text{ m}^2$$ - Total area: $$A = 30 + 6 = 36\text{ m}^2$$ --- ### b) Teardrop-shaped figure with a semicircle on top (diameter 8 cm) and two equal slanted sides 10 cm each meeting at a point below. 6. **Perimeter Calculation:** - The perimeter includes the semicircle arc plus the two slanted sides. - Semicircle circumference: $$C = \pi \times d = \pi \times 8 = 8\pi$$ - Semicircle arc length is half the circumference: $$L = \frac{8\pi}{2} = 4\pi \approx 12.6\text{ cm}$$ - Add the two slanted sides: $$P = 12.6 + 10 + 10 = 32.6\text{ cm}$$ 7. **Area Calculation:** - Area of semicircle: $$A_{semi} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (4)^2 = 8\pi \approx 25.1\text{ cm}^2$$ - Area of triangle formed by two slanted sides and base 8 cm: - Use Heron's formula: $$s = \frac{10 + 10 + 8}{2} = 14$$ $$A_{tri} = \sqrt{14(14-10)(14-10)(14-8)} = \sqrt{14 \times 4 \times 4 \times 6} = \sqrt{1344} \approx 36.7\text{ cm}^2$$ - Total area: $$A = 25.1 + 36.7 = 61.8\text{ cm}^2$$ --- **Final answers:** - a) Perimeter = 28 m, Area = 36 m² - b) Perimeter = 32.6 cm, Area = 61.8 cm²