To determine the relationship between the lines CD and FG, we need to find the slopes of both lines and compare them.
Finding the slope of line CD: The points on line CD are (-2, 4) and (0, -4).
The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates for line CD:
\[ m_{CD} = \frac{-4 - 4}{0 - (-2)} = \frac{-8}{2} = -4 \]
Finding the slope of line FG: The points on line FG are (-4, 0) and (4, 2).
Using the slope formula again:
\[ m_{FG} = \frac{2 - 0}{4 - (-4)} = \frac{2}{4 + 4} = \frac{2}{8} = \frac{1}{4} \]
Comparing the slopes:
- The slope of line CD \(m_{CD} = -4\)
- The slope of line FG \(m_{FG} = \frac{1}{4}\)
Now, we check if they are negative reciprocals:
To be negative reciprocals, the product of the two slopes should equal -1:
\[ m_{CD} \cdot m_{FG} = (-4) \cdot \left(\frac{1}{4}\right) = -1 \]
Since the product of the slopes is -1, this means the two lines are indeed perpendicular.
Thus, the correct statement is:
They are perpendicular because their slopes are negative reciprocals.