Question

On a coordinate plane, 4 lines are shown. Line A B goes through (negative 4, negative 2) and (4, 2). Line C D goes through (negative 4, 0) and (4, negative 4). Line F G goes through (negative 3, negative 3) and (0, 3). Line H J goes through (negative 1, 3) and (1, negative 1).
Which line is perpendicular to a line that has a slope of One-half?

line AB
line CD
line FG
line HJ

Answer 1

To find which line is perpendicular to a line that has a slope of \( \frac{1}{2} \), we need to determine the slopes of the given lines and check if their slopes are the negative reciprocal of \( \frac{1}{2} \).

The negative reciprocal of \( \frac{1}{2} \) is \( -2 \) (since the product of the slopes of two perpendicular lines is \(-1\)).

Now, let's find the slopes of the given lines.

**Line AB:**
Points: \( (-4, -2) \) and \( (4, 2) \)
Slope \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-2)}{4 - (-4)} = \frac{4}{8} = \frac{1}{2} \)

**Line CD:**
Points: \( (-4, 0) \) and \( (4, -4) \)
Slope \( m = \frac{-4 - 0}{4 - (-4)} = \frac{-4}{8} = -\frac{1}{2} \)

**Line FG:**
Points: \( (-3, -3) \) and \( (0, 3) \)
Slope \( m = \frac{3 - (-3)}{0 - (-3)} = \frac{6}{3} = 2 \)

**Line HJ:**
Points: \( (-1, 3) \) and \( (1, -1) \)
Slope \( m = \frac{-1 - 3}{1 - (-1)} = \frac{-4}{2} = -2 \)

Now we compare the calculated slopes to see which is \( -2 \):

- Line AB has a slope of \( \frac{1}{2} \)
- Line CD has a slope of \( -\frac{1}{2} \)
- Line FG has a slope of \( 2 \)
- Line HJ has a slope of \( -2 \)

Thus, **Line HJ**, having a slope of \( -2 \), is perpendicular to a line with a slope of \( \frac{1}{2} \).