Consider the following four points: \( \begin{array}{ll}\text { A }(12,14) & \text { C }(24,-22) \\ \text { B }(16,-25) & \text { D }(18,8) \\ \text { Will the lines } A B \text { and CD intersect } 8 \text { ? } \\ \text { a) yes } \\ \text { b) no }\end{array} \)

To determine whether the lines \( AB \) and \( CD \) intersect at the point \( (8, y) \), we need to find the equations of both lines and then check if they yield the same \( y \)-value when \( x = 8 \). ### Step 1: Find the equation of line \( AB \) The points \( A(12, 14) \) and \( B(16, -25) \) can be used to find the slope \( m_{AB} \) of line \( AB \): \[ m_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{-25 - 14}{16 - 12} = \frac{-39}{4} \] Now, we can use point-slope form to find the equation of line \( AB \): \[ y - y_A = m_{AB}(x - x_A) \] Substituting \( A(12, 14) \): \[ y - 14 = \frac{-39}{4}(x - 12) \] Now, let's simplify this equation: \[ y - 14 = \frac{-39}{4}x + \frac{39 \cdot 12}{4} \] \[ y - 14 = \frac{-39}{4}x + 117 \] \[ y = \frac{-39}{4}x + 131 \] ### Step 2: Find the equation of line \( CD \) Next, we find the slope \( m_{CD} \) of line \( CD \) using points \( C(24, -22) \) and \( D(18, 8) \): \[ m_{CD} = \frac{y_D - y_C}{x_D - x_C} = \frac{8 - (-22)}{18 - 24} = \frac{30}{-6} = -5 \] Using point-slope form for line \( CD \): \[ y - y_C = m_{CD}(x - x_C) \] Substituting \( C(24, -22) \): \[ y - (-22) = -5(x - 24) \] \[ y + 22 = -5x + 120 \] \[ y = -5x + 98 \] ### Step 3: Check for intersection at \( x = 8 \) Now we will substitute \( x = 8 \) into both equations to find the corresponding \( y \)-values. For line \( AB \): \[ y_{AB} = \frac{-39}{4}(8) + 131 \] Calculating this: \[ y_{AB} = \frac{-312}{4} + 131 = -78 + 131 = 53 \] For line \( CD \): \[ y_{CD} = -5(8) + 98 \] Calculating this: \[ y_{CD} = -40 + 98 = 58 \] ### Conclusion Since \( y_{AB} = 53 \) and \( y_{CD} = 58 \), the lines \( AB \) and \( CD \) do not intersect at \( x = 8 \). Thus, the answer is: **b) no**