1. **State the problem:** We have points A(0,5), B(-4,0), and C(4,0) representing parts of a swing frame. We need to find: (i) The coordinates of A, B, and C. (ii) The total length of metal bars AB and AC. (iii) The slopes of AB and AC. (iv) Whether AB is perpendicular to AC. 2. **Coordinates:** (i) The coordinates are given as: - A = (0, 5) - B = (-4, 0) - C = (4, 0) 3. **Length of metal bars:** Use the distance formula between two points $ (x_1,y_1) $ and $ (x_2,y_2) $: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$ 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: $$ |AB| + |AC| = \sqrt{41} + \sqrt{41} = 2\sqrt{41} $$ Calculate decimal value: $$ 2\sqrt{41} \approx 2 \times 6.4031 = 12.8062 $$ Rounded to one decimal place: $$ 12.8 $$ meters 4. **Slopes of AB and AC:** 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: $$ m_{AC} = \frac{0 - 5}{4 - 0} = \frac{-5}{4} = -1.25 $$ 5. **Are AB and AC perpendicular?** Two lines are perpendicular if the product of their slopes is $-1$: $$ m_{AB} \times m_{AC} = 1.25 \times (-1.25) = -1.5625 \neq -1 $$ Therefore, AB is **not** perpendicular to AC. **Final answers:** (i) A(0,5), B(-4,0), C(4,0) (ii) Total length = 12.8 meters (iii) $m_{AB} = 1.25$, $m_{AC} = -1.25$ (iv) AB is not perpendicular to AC because the product of slopes is not $-1$.