Question

determine whether the following statement makes sense or does not make sense, and explain your reasoning. the slope of one line was computed to be $-\frac{4}{9}$ and slope of the second line to be $-\frac{9}{4}$, so the lines must be perpendicular. select the correct choice below and, if necessary, fill in the answer box to complete your choice. \\(\bigcirc\\) a. the statement does not make sense because if two lines are perpendicular, then their slopes are negative reciprocals of each other. thus, if a line has slope $-\frac{4}{9}$, a perpendicular line has slope \\(\bigcirc\\) b. the statement makes sense because one line is perpendicular to another line if its slope is the reciprocal of the slope of the other line. \\(\bigcirc\\) c. the statement does not make sense because perpendicular lines have the same slope. \\(\bigcirc\\) d. the statement makes sense because the product of the slopes of two perpendicular lines is 1.

Explanation:

Step1: Recall perpendicular slope rule

Two lines are perpendicular if the product of their slopes is $-1$, meaning slopes are negative reciprocals.

Step2: Calculate negative reciprocal of $-\frac{4}{9}$

The negative reciprocal is $\frac{9}{4}$, since $-\frac{4}{9} \times \frac{9}{4} = -1$.

Step3: Compare with given second slope

The given second slope is $-\frac{9}{4}$, which is not the negative reciprocal of $-\frac{4}{9}$.

Answer:

A. The statement does not make sense because if two lines are perpendicular, then their slopes are negative reciprocals of each other. Thus, if a line has slope $-\frac{4}{9}$, a perpendicular line has slope $\frac{9}{4}$