Questions: 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 -3/4 and slope of the second line to be -4/3, so the lines must be perpendicular. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The statement makes sense because the product of the slopes of two perpendicular lines is 1. B. 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 -3/4, a perpendicular line has slope C. 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 D. The statement does not make sense because perpendicular lines have the same slope.

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 -3/4 and slope of the second line to be -4/3, so the lines must be perpendicular.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The statement makes sense because the product of the slopes of two perpendicular lines is 1.
B. 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 -3/4, a perpendicular line has slope
C. 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
D. The statement does not make sense because perpendicular lines have the same slope.

Transcript text: 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 $-\frac{3}{4}$ and slope of the second line to be $-\frac{4}{3}$, so the lines must be perpendicular. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The statement makes sense because the product of the slopes of two perpendicular lines is 1. B. 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 $-\frac{3}{4}$, a perpendicular line has slope $\square$ C. 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 D. The statement does not make sense because perpendicular lines have the same slope.

Solution

Solution Steps

To determine if the statement makes sense, we need to recall the property of perpendicular lines: two lines are perpendicular if the product of their slopes is -1. Given the slopes $-\frac{3}{4}$ and $-\frac{4}{3}$, we can calculate their product to verify if it equals -1.

Step 1: Identify the Slopes

The slopes of the two lines are given as: \[ m_1 = -\frac{3}{4} \quad \text{and} \quad m_2 = -\frac{4}{3} \]

Step 2: Calculate the Product of the Slopes

We calculate the product of the slopes: \[ m_1 \cdot m_2 = \left(-\frac{3}{4}\right) \cdot \left(-\frac{4}{3}\right) = \frac{3 \cdot 4}{4 \cdot 3} = 1 \]

Step 3: Determine Perpendicularity

For two lines to be perpendicular, the product of their slopes must equal $-1$. Since we found that: \[ m_1 \cdot m_2 = 1 \neq -1 \] the lines are not perpendicular.