Which line is perpendicular to a line that has a slope of $-\frac{5}{6}$?
A. line JK
B. line LM
C. line NO
D. line $P Q$
Answer (2)
To find a line perpendicular to a line with a slope of − 6 5 :
Recall that perpendicular lines have slopes that are negative reciprocals of each other.
Calculate the negative reciprocal of − 6 5 , which is 5 6 .
Identify the line with the slope 5 6 .
The line PQ is perpendicular to the given line, since it has a slope of 5 6 .
Explanation
Understanding the problem The problem asks us to find a line that is perpendicular to a line with a given slope. Let's recall the relationship between the slopes of perpendicular lines.
Slopes of perpendicular lines Two lines are perpendicular if the product of their slopes is -1. Equivalently, if a line has slope m , a line perpendicular to it has slope − m 1 . This is called the negative reciprocal of the slope.
Finding the negative reciprocal The given line has a slope of − 6 5 . To find the slope of a line perpendicular to it, we need to find the negative reciprocal of − 6 5 .
Calculating the negative reciprocal The negative reciprocal of − 6 5 is
− ( − 6 5 ) 1 = 5 6
Identifying the perpendicular line Therefore, a line perpendicular to the given line has a slope of 5 6 . We need to identify which of the given lines (JK, LM, NO, PQ) has this slope.
Final Answer The line PQ is the line that has a slope of 5 6 .
Examples
In architecture, ensuring walls are perpendicular is crucial for structural integrity. If a design requires a wall to be perpendicular to another with a known slope, architects use the principle of negative reciprocals to calculate the necessary slope for the new wall, ensuring it stands correctly and safely.
A line is perpendicular to a line with a slope of − 6 5 if it has a slope of 5 6 . After assessing the options provided, the correct answer is line PQ. This line has the requisite slope, making it perpendicular to the original line.
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