Answer to Question #131431 in Geometry for sage

Question

A software designer is mapping the streets for a new racing game. All of the streets are depicted as either perpendicular or parallel lines. The equation of the lane passing through A and B is -7x + 3y = -21.5. What is the equation of the central street PQ?

Answer 1

The equation of the lane passing through A and B is -7x + 3y = -21.5

this could be written as y=$\frac{7}{3}$x​−$\frac{21.5}{3}$

By comparing with the standard form y=mx+c

The slope m=$\frac{7}{3}$

slope of this line is 7/3

central street PQ will be perpendicular to the lane passing through A and B

Product of Slope of perpendicular lines will be -1

m₁m₂₌-1

Slope of perpendicular line = $\frac{-1}{slope\;of\;parallel\;line}$ =$\frac{-1}{\frac{7}{3}}$ =$\frac{-3}{\;7}$

This is the slope of the line.

Thus the equation of line will be

y=mx+c

y=$\frac{-3}{7}$x+c

7y= -3x+7c

7y+3x=7c

Dividing by 2

3.5y+1.5x=3.5c

To find c with just this information is impossible.

We will have to cross check the answer with the options (if options are given) i.

In the question no other information is given, so if there are options we have to check for the line with slope as -3/7

OR in a similar question a figure is found

The lane PQ passes through the point (7,6) in the figure.

Hence the equation of line could be found out using point slope form of a line.

$y-y_1=m(x-x_1)$

slope =$\frac{-3}{7}$

(7,6)

y-6=$\frac{-3}{7}$ (x-7)

7(y-6)=-3(x-7)

7y-42=-3x+21

7y+3x=21+42

7y+3x=63 is the equation of central lane

Dividing by 2

3.5y+1.5x=31.5

OR -3.5y-1.5x=-31.5 is also the equation