On a coordinate plane, 2 lines are shown. Line H J has points (negative 4, negative 2) and (0, 4). Line F G has points (negative 4, 1) and (0, negative 2).

Question

Which statement best explains the relationship between lines FG and HJ?

They are perpendicular because their slopes are equal.
They are perpendicular because their slopes are negative reciprocals.
They are not perpendicular because their slopes are equal.
They are not perpendicular because their slopes are not negative reciprocals.

Answer 1

To determine the relationship between lines FG and HJ, we first need to find the slopes of both lines.

Line HJ: The points are \((-4, -2)\) and \( (0, 4)\).

The slope \(m\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting in the coordinates: \[ m_{HJ} = \frac{4 - (-2)}{0 - (-4)} = \frac{4 + 2}{0 + 4} = \frac{6}{4} = \frac{3}{2} \]

Line FG: The points are \((-4, 1)\) and \( (0, -2)\).

Using the same slope formula: \[ m_{FG} = \frac{-2 - 1}{0 - (-4)} = \frac{-3}{4} \]

Now we compare the slopes:

• The slope of line HJ is \(\frac{3}{2}\)
• The slope of line FG is \(\frac{-3}{4}\)

To check if the lines are perpendicular, we consider whether the slopes are negative reciprocals of each other. The negative reciprocal of \(\frac{3}{2}\) is \(-\frac{2}{3}\), but since \(\frac{-3}{4} \neq -\frac{2}{3}\), the slopes are not negative reciprocals.

Conclusion: The lines are not perpendicular because their slopes are not negative reciprocals.

Thus, the correct statement is: They are not perpendicular because their slopes are not negative reciprocals.