1. **Problem statement:** Find the total length of metal bar needed to make the part of the swing formed by points A(0,5), B(-4,0), and C(4,0). Then find the slopes of AB and AC, and determine if AB is perpendicular to AC. 2. **Length of metal bar (ii):** We need to find the lengths of AB and AC and add them. Length formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is: $$\text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Calculate $AB$: $$AB = \sqrt{(-4 - 0)^2 + (0 - 5)^2} = \sqrt{(-4)^2 + (-5)^2} = \sqrt{16 + 25} = \sqrt{41}$$ Calculate $AC$: $$AC = \sqrt{(4 - 0)^2 + (0 - 5)^2} = \sqrt{4^2 + (-5)^2} = \sqrt{16 + 25} = \sqrt{41}$$ Total length: $$AB + AC = \sqrt{41} + \sqrt{41} = 2\sqrt{41}$$ Approximate value: $$2 \times 6.4031 = 12.8062 \approx 12.8$$ meters (to one decimal place). 3. **Slopes of AB and AC (iii):** Slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Slope of $AB$: $$m_{AB} = \frac{0 - 5}{-4 - 0} = \frac{-5}{-4} = \frac{5}{4} = 1.25$$ Slope of $AC$: $$m_{AC} = \frac{0 - 5}{4 - 0} = \frac{-5}{4} = -1.25$$ 4. **Are AB and AC perpendicular? (iv):** Two lines are perpendicular if the product of their slopes is $-1$. Calculate: $$m_{AB} \times m_{AC} = 1.25 \times (-1.25) = -1.5625 \neq -1$$ **Answer:** No, AB is not perpendicular to AC. **Reason:** The product of their slopes is not $-1$, so the lines are not perpendicular.