Which line is perpendicular to a line that has a slope of $-\frac{1}{3}$?
A. line MN
B. line AB
C. line EF
D. line JK
Answer (2)
The slope of a line that is perpendicular to a line with a slope of − 3 1 is 3. Therefore, to identify the correct option among lines MN, AB, EF, and JK, look for the one with a slope of 3. Without additional information about the slopes, we cannot directly select an option.
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The problem requires finding a line perpendicular to a line with a slope of − 3 1 .
Recall that the product of the slopes of perpendicular lines is -1.
Calculate the slope of the perpendicular line: m 2 = 3 .
Identify the line with a slope of 3 as the perpendicular line.
Explanation
Problem Analysis The problem asks us to identify which line is perpendicular to a line with a slope of − 3 1 . We need to remember the relationship between the slopes of perpendicular lines.
Perpendicular Lines Condition Two lines are perpendicular if the product of their slopes is -1. Let m 1 be the slope of the given line and m 2 be the slope of the line perpendicular to it. Then, we have: m 1 m 2 = − 1
Substitute the Given Slope We are given that m 1 = − 3 1 . We need to find m 2 such that: − 3 1 ⋅ m 2 = − 1
Solve for the Perpendicular Slope To solve for m 2 , we can multiply both sides of the equation by -3: m 2 = − 1 ⋅ ( − 3 ) = 3
Identify the Correct Line Therefore, the slope of the line perpendicular to the given line is 3. Now we need to check which of the given lines (MN, AB, EF, JK) has a slope of 3.
Final Answer The line with a slope of 3 is the line perpendicular to the given line. Without knowing the slopes of lines MN, AB, EF, and JK, we can only say that the line with slope 3 is perpendicular to the line with slope − 3 1 .
Examples
In architecture, ensuring walls are perpendicular is crucial for structural integrity. If a design requires a wall to be perpendicular to a wall with a known slope, architects use the principle that the product of the slopes of perpendicular lines must equal -1. This ensures precise alignment and stability in construction.
