A triangle has vertices \( A(2,-2), B(6,4) \) and \( C(10,1) \). Determine the equation of the sides of the triangle. A parallelogram has vertices \( A(2,1), B(5,4), C(9,4) \) and \( D(6,1) \). Determine the equation of line: \( \begin{array}{llll}\text { a) } A B & \text { b) } B C & \text { c) } C D & \text { d) } A D\end{array} \)

To find the equations of the sides of the triangle and the parallelogram, we will use the formula for the equation of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\): \[ y - y_1 = m(x - x_1) \] where \(m\) is the slope calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] ### Step 1: Determine the equations of the sides of the triangle \(ABC\) **Vertices of the triangle:** - \(A(2, -2)\) - \(B(6, 4)\) - \(C(10, 1)\) #### Side AB: 1. Calculate the slope \(m_{AB}\): \[ m_{AB} = \frac{4 - (-2)}{6 - 2} = \frac{6}{4} = \frac{3}{2} \] 2. Use point \(A(2, -2)\) to find the equation: \[ y - (-2) = \frac{3}{2}(x - 2) \] Simplifying: \[ y + 2 = \frac{3}{2}x - 3 \implies y = \frac{3}{2}x - 5 \] #### Side BC: 1. Calculate the slope \(m_{BC}\): \[ m_{BC} = \frac{1 - 4}{10 - 6} = \frac{-3}{4} \] 2. Use point \(B(6, 4)\) to find the equation: \[ y - 4 = -\frac{3}{4}(x - 6) \] Simplifying: \[ y - 4 = -\frac{3}{4}x + \frac{18}{4} \implies y = -\frac{3}{4}x + \frac{34}{4} \implies y = -\frac{3}{4}x + 8.5 \] #### Side CA: 1. Calculate the slope \(m_{CA}\): \[ m_{CA} = \frac{-2 - 1}{2 - 10} = \frac{-3}{-8} = \frac{3}{8} \] 2. Use point \(C(10, 1)\) to find the equation: \[ y - 1 = \frac{3}{8}(x - 10) \] Simplifying: \[ y - 1 = \frac{3}{8}x - \frac{30}{8} \implies y = \frac{3}{8}x - \frac{22}{8} \implies y = \frac{3}{8}x - 2.75 \] ### Step 2: Determine the equations of the sides of the parallelogram \(ABCD\) **Vertices of the parallelogram:** - \(A(2, 1)\) - \(B(5, 4)\) - \(C(9, 4)\) - \(D(6, 1)\) #### Side AB: 1. Calculate the slope \(m_{AB}\): \[ m_{AB} = \frac{4 - 1}{5 - 2} = \frac{3}{3} = 1 \] 2. Use point \(A(2, 1)\) to find the equation: \[ y - 1 = 1(x - 2) \implies y = x - 1 \] #### Side BC: 1. Calculate the slope \(m_{BC}\): \[ m_{BC} = \frac{4 - 4}{9 - 5} = \frac{0}{4} = 0 \] 2. The equation is simply: \[ y = 4 \] #### Side CD: 1. Calculate the slope \(m_{CD}\): \[ m_{CD} = \frac{1 - 4}{6 - 9} = \frac{-3}{-3} = 1 \] 2. Use point \(C(9, 4)\) to find the equation: \[ y - 4 = 1(x - 9) \implies y = x - 5 \] #### Side AD: 1. Calculate the slope \(m_{AD}\): \[ m_{AD} = \frac{1 - 1}{2 - 6} = \frac{0}{-4} = 0 \] 2. The equation is simply: \[ y = 1 \] ### Summary of Equations - **Triangle \(ABC\)**: - Side \(AB\): \(y = \frac{3}{2}x - 5\) - Side \(BC\): \(y = -\frac{3}{4}x + 8.5\) - Side \(CA\): \(y = \frac{3}{8}x - 2.75\) - **Parallelogram \(ABCD\)**: - Side \(AB\): \(y = x - 1\) - Side \(BC\): \(y = 4\) - Side \(CD\): \(y = x - 5\) - Side \(AD\): \(y = 1\)