1. **Problem statement:** We have a roof support made of four wooden lengths: AB, AC, BC, and MC. Given: - AC = BC = 9 m - AB = 12 m - Angle AMC = 90° (meaning MC is perpendicular to AB) We need to find the total cost of wood Lewis needs to buy, knowing the wood costs 21.50 per metre and each length must be a whole number of metres. 2. **Understanding the problem:** - Triangle ABC is isosceles with AC = BC = 9 m and base AB = 12 m. - M is the foot of the perpendicular from C to AB, so triangle AMC is right angled at M. 3. **Find length MC:** Since M is the foot of the perpendicular from C to AB, and AB = 12 m, M divides AB into AM and MB. Because triangle ABC is isosceles with AC = BC, M is the midpoint of AB. So, $$AM = MB = \frac{12}{2} = 6 \text{ m}$$ 4. **Use Pythagoras theorem in triangle AMC:** $$AC^2 = AM^2 + MC^2$$ $$9^2 = 6^2 + MC^2$$ $$81 = 36 + MC^2$$ $$MC^2 = 81 - 36 = 45$$ $$MC = \sqrt{45} = 3\sqrt{5} \approx 6.708 \text{ m}$$ 5. **Calculate total length of wood:** - AB = 12 m - AC = 9 m - BC = 9 m - MC \approx 6.708 m Since lengths must be whole numbers, round MC up to 7 m. Total length = 12 + 9 + 9 + 7 = 37 m 6. **Calculate total cost:** $$\text{Cost} = 37 \times 21.50 = 795.5$$ **Final answer:** The total cost of the wood Lewis needs to buy is 795.5.