Questions: 12 In trapezium ABCD, ABD and ACD are right-angled triangles. a) Find the area of triangle ABD. b) Hence find the height of the trapezium. c) Find the area of the trapezium. d) Find the area of triangle ABC. e) Show that triangle AED is 108 m² larger than triangle BEC.

△ Find the area of triangle ABD. ○ Calculate area using base and height ☼ The base of triangle ABD is AD = 15 m and its height is AB = 25 m. The area is \( \frac{1}{2} \times 15 \, \text{m} \times 25 \, \text{m} = 187.5 \, \text{m}^2 \). ✧ The area of triangle ABD is \( 187.5 \, \text{m}^2 \).

△ Find the height of the trapezium. ○ Identify the height ☼ The height of the trapezium is the perpendicular distance between the parallel sides AB and CD, which is the length of BC. Given BC = 7 m. ✧ The height of the trapezium is 7 m.

△ Find the area of the trapezium ABCD. ○ Calculate area using trapezium formula ☼ The area of a trapezium is given by \( \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height} \). Thus, \( \frac{1}{2} \times (25 \, \text{m} + 7 \, \text{m}) \times 7 \, \text{m} = 112 \, \text{m}^2 \). ✧ The area of the trapezium ABCD is \( 112 \, \text{m}^2 \).

△ Find the area of triangle ABC. ○ Calculate area using base and height ☼ The base of triangle ABC is AB = 25 m, and the height is BC = 7 m. The area is \( \frac{1}{2} \times 25 \, \text{m} \times 7 \, \text{m} = 87.5 \, \text{m}^2 \). ✧ The area of triangle ABC is \( 87.5 \, \text{m}^2 \).

△ Show that triangle AED is 108 m² larger than triangle BEC. ○ Calculate areas of AED and BEC ☼ The area of triangle ACD is \( 112 \, \text{m}^2 - 87.5 \, \text{m}^2 = 24.5 \, \text{m}^2 \). Given AE = EC = 20 m, the area of AED is \( \frac{20}{40} \times 24.5 \, \text{m}^2 = 12.25 \, \text{m}^2 \) and the area of BEC is \( \frac{20}{40} \times 87.5 \, \text{m}^2 = 43.75 \, \text{m}^2 \). The difference is \( 43.75 \, \text{m}^2 - 12.25 \, \text{m}^2 = 31.5 \, \text{m}^2 \), not 108 m². ✧ The comparison does not hold; the difference is \( 31.5 \, \text{m}^2 \). ☺ a. Area(ABD) = 187.5 m² b. Height of trapezium = 7 m c. Area(ABCD) = 112 m² d. Area(ABC) = 87.5 m² e. The prompt requests a proof of comparison that doesn't hold based on the given diagram's measurements. Please verify the prompt.