Answer :
To determine the equation of the central street \( PQ \), which is perpendicular to the lane passing through points \( A \) and \( B \), we need to go through a series of steps. Let's begin.
### Step 1: Identify the Slope of the Given Lane
The equation of the lane passing through \( A \) and \( B \) is given by:
[tex]\[ -7x + 3y = -21.5 \][/tex]
We need to determine the slope of this line. First, we convert this equation to the slope-intercept form, \( y = mx + b \), where \( m \) is the slope.
Let's isolate \( y \):
[tex]\[ 3y = 7x - 21.5 \][/tex]
[tex]\[ y = \frac{7}{3}x - \frac{21.5}{3} \][/tex]
From this equation, we can see that the slope \( m \) of the line \( AB \) is:
[tex]\[ m = \frac{7}{3} \][/tex]
### Step 2: Determine the Slope of the Perpendicular Line
The central street \( PQ \) should be perpendicular to \( AB \). The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.
The slope of \( AB \) is \( \frac{7}{3} \). Therefore, the slope of \( PQ \) (perpendicular to \( AB \)) is:
[tex]\[ -\frac{1}{\frac{7}{3}} = -\frac{3}{7} \][/tex]
### Step 3: Form the Equation of the Perpendicular Line
Now we need the equation of the line with the slope \( -\frac{3}{7} \). We can start by using the slope-intercept form \( y = mx + b \), then convert it to the general form \( Ax + By = C \).
The line equation with slope \( -\frac{3}{7} \):
[tex]\[ y = -\frac{3}{7}x + b \][/tex]
Rearrange the equation to the form \( Ax + By = C \):
[tex]\[ 7y = -3x + 7b \][/tex]
[tex]\[ 3x + 7y = 7b \][/tex]
This is the general form of the equation of the central street \( PQ \).
### Step 4: Identify the Correct Option
To match this form with the provided options, compare:
[tex]\[ 3x + 7y = C \][/tex]
The constant \( 7b \) in our general form represents the value on the right-hand side. Therefore, the correct equation from the given choices that fits this form is:
[tex]\[ 3x + 7y = 63 \][/tex]
Thus, the correct equation for the central street \( PQ \) is:
B. [tex]\( 3x + 7y = 63 \)[/tex]
### Step 1: Identify the Slope of the Given Lane
The equation of the lane passing through \( A \) and \( B \) is given by:
[tex]\[ -7x + 3y = -21.5 \][/tex]
We need to determine the slope of this line. First, we convert this equation to the slope-intercept form, \( y = mx + b \), where \( m \) is the slope.
Let's isolate \( y \):
[tex]\[ 3y = 7x - 21.5 \][/tex]
[tex]\[ y = \frac{7}{3}x - \frac{21.5}{3} \][/tex]
From this equation, we can see that the slope \( m \) of the line \( AB \) is:
[tex]\[ m = \frac{7}{3} \][/tex]
### Step 2: Determine the Slope of the Perpendicular Line
The central street \( PQ \) should be perpendicular to \( AB \). The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.
The slope of \( AB \) is \( \frac{7}{3} \). Therefore, the slope of \( PQ \) (perpendicular to \( AB \)) is:
[tex]\[ -\frac{1}{\frac{7}{3}} = -\frac{3}{7} \][/tex]
### Step 3: Form the Equation of the Perpendicular Line
Now we need the equation of the line with the slope \( -\frac{3}{7} \). We can start by using the slope-intercept form \( y = mx + b \), then convert it to the general form \( Ax + By = C \).
The line equation with slope \( -\frac{3}{7} \):
[tex]\[ y = -\frac{3}{7}x + b \][/tex]
Rearrange the equation to the form \( Ax + By = C \):
[tex]\[ 7y = -3x + 7b \][/tex]
[tex]\[ 3x + 7y = 7b \][/tex]
This is the general form of the equation of the central street \( PQ \).
### Step 4: Identify the Correct Option
To match this form with the provided options, compare:
[tex]\[ 3x + 7y = C \][/tex]
The constant \( 7b \) in our general form represents the value on the right-hand side. Therefore, the correct equation from the given choices that fits this form is:
[tex]\[ 3x + 7y = 63 \][/tex]
Thus, the correct equation for the central street \( PQ \) is:
B. [tex]\( 3x + 7y = 63 \)[/tex]
