To determine the missing coordinate of point C in line CD such that line AB and line CD are perpendicular, we first need to find the slopes of both lines and set the product of their slopes equal to -1 (the condition for perpendicular lines).
Step 1: Calculate the slope of line AB
Let Point 1 be \( A(4, 6) \) and Point 2 be \( B(3, 8) \).
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For line AB: \[ m_{AB} = \frac{8 - 6}{3 - 4} = \frac{2}{-1} = -2 \]
Step 2: Calculate the slope of line CD
Let Point 1 be \( C(1, 5) \) and Point 2 be \( D(x, 4) \) where \( x \) is the missing x-coordinate.
Using the slope formula again, we have: \[ m_{CD} = \frac{4 - 5}{x - 1} = \frac{-1}{x - 1} \]
Step 3: Set up the equation for the slopes to be perpendicular
The lines are perpendicular if: \[ m_{AB} \cdot m_{CD} = -1 \] Substituting in the slopes we found: \[ -2 \cdot \frac{-1}{x - 1} = -1 \]
Simplifying the equation: \[ \frac{2}{x - 1} = -1 \]
Step 4: Solve for \( x \)
Multiplying both sides by \( x - 1 \): \[ 2 = -1(x - 1) \] \[ 2 = -x + 1 \] Adding \( x \) to both sides: \[ x + 2 = 1 \] Subtracting 2 from both sides: \[ x = 1 - 2 = -1 \]
Thus, the missing coordinate for point D in line CD is: \[ \boxed{-1} \]