Line m has a y-intercept of c and a slope of [tex]$\frac{p}{q}$[/tex], where [tex]$p\ \textgreater \ 0, q\ \textgreater \ 0$[/tex], and [tex]$p \neq q$[/tex]. What is the slope of a line that is perpendicular to line m?
A. [tex]$-\frac{p}{q}$[/tex]
B. [tex]$\frac{p}{q}$[/tex]
C. [tex]$-\frac{q}{p}$[/tex]
D. [tex]$\frac{q}{p}$[/tex]
Answer (2)
The slope of line m is q p .
The slope of a line perpendicular to line m is the negative reciprocal of the slope of line m.
The negative reciprocal of q p is − p q .
Therefore, the slope of a line perpendicular to line m is − p q .
Explanation
Understanding the Problem We are given that line m has a slope of q p , where 0"> p > 0 and 0"> q > 0 . We need to find the slope of a line that is perpendicular to line m .
Key Concept The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line.
Finding the Perpendicular Slope The slope of line m is q p . Therefore, the slope of a line perpendicular to line m is the negative reciprocal of q p , which is − p q .
Final Answer Thus, the slope of a line perpendicular to line m is − p q .
Examples
Imagine you're designing a roof. The slope of the roof is q p . To ensure proper water runoff, you need to determine the slope of a supporting beam that is perpendicular to the roof. This problem demonstrates how to calculate the slope of that supporting beam, ensuring it's at a right angle to the roof for optimal structural integrity.
The slope of a line perpendicular to line m, which has a slope of q p , is − p q . This is because the slope of a perpendicular line is the negative reciprocal of the original slope. Therefore, the correct answer is option C: − p q .
;
