Question

which line is perpendicular to a line that has a slope of $-\frac{1}{3}$?
○ line mn
○ line ab
○ line ef
○ line jk

Explanation:

Step1: Recall perpendicular slope rule

The slope of a line perpendicular to a line with slope \( m \) is the negative reciprocal, i.e., \( -\frac{1}{m} \). For \( m = -\frac{1}{3} \), the perpendicular slope is \( 3 \) (since \( -\frac{1}{-\frac{1}{3}} = 3 \)).

Step2: Calculate slopes of each line

• Line MN: Points \( M(-1, 4) \) and \( N(2, -5) \). Slope \( m_{MN} = \frac{-5 - 4}{2 - (-1)} = \frac{-9}{3} = -3 \).
• Line AB: Points \( A(-2, 2) \) and \( B(3, 0) \). Slope \( m_{AB} = \frac{0 - 2}{3 - (-2)} = \frac{-2}{5} = -\frac{2}{5} \).
• Line EF: Points \( E(1, -3) \) and \( F(2, 0) \). Slope \( m_{EF} = \frac{0 - (-3)}{2 - 1} = \frac{3}{1} = 3 \).
• Line JK: Points \( J(-3, -4) \) and \( K(3, -2) \). Slope \( m_{JK} = \frac{-2 - (-4)}{3 - (-3)} = \frac{2}{6} = \frac{1}{3} \).

Step3: Identify the line with slope 3

Line EF has a slope of 3, which is the negative reciprocal of \( -\frac{1}{3} \), so it is perpendicular.

Answer:

line EF