Which line is perpendicular to a line that has a slope of $-\frac{5}{6}$?
line JK
line LM
line NO
line PQ
Answer (1)
The problem requires finding 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 slope of the perpendicular line: m 2 = − m 1 1 = 5 6 .
Identify the line (JK, LM, NO, or PQ) that has a slope of 5 6 : 5 6 .
Explanation
Analyze the problem and given data The problem asks us to find a line that is perpendicular to a given line with a slope of − 6 5 . Remember that two lines are perpendicular if the product of their slopes is -1. Let's denote the slope of the given line as m 1 = − 6 5 and the slope of the line we are looking for as m 2 .
State the condition for perpendicularity The condition for perpendicularity is that the product of the slopes is -1, which means m 1 ⋅ m 2 = − 1 . We can solve for m 2 by dividing both sides of the equation by m 1 : m 2 = m 1 − 1
Calculate the slope of the perpendicular line Substitute the given slope m 1 = − 6 5 into the equation to find m 2 :
m 2 = − 6 5 − 1 = − 1 ⋅ ( − 5 6 ) = 5 6 So, the slope of the line perpendicular to the given line is 5 6 .
Determine the line with the calculated slope Now we need to determine which of the lines JK, LM, NO, or PQ has a slope of 5 6 . Without additional information about the slopes of these lines, we cannot definitively choose one. However, if we assume that one of these lines indeed has a slope of 5 6 , then that would be the correct answer. Therefore, the slope of the perpendicular line is 5 6 .
Final Answer Without knowing the slopes of lines JK, LM, NO, and PQ, we can't definitively say which one is perpendicular. However, we know the perpendicular slope must be 5 6 .
Examples
Understanding perpendicular slopes is crucial in architecture and construction. When designing buildings, ensuring walls are perpendicular to the ground or that different sections of a structure meet at right angles is essential for stability and safety. For example, calculating the slope of a roof and ensuring it's perpendicular to a supporting beam involves using the concept of negative reciprocal slopes to guarantee structural integrity.
