Answer to Question #210517 in Geometry for doug

Question

A and B is -7x + 3y = -21.5. What is the equation of the central street PQ?

Answer 1

the given line is

$-7x+3y=-21.5\\ 3y=7x-21.5\\ y=\frac{7}{3}x-\frac{21.5}{3}$

here coefficient of x is 7/3 so the slope of the given line is

$m_1=\frac{7}{3}$

now the line PQ should be either parallel or perpendicular 

let the slope of the line PQ is  m₂

so when m₁=m₂  then both lines are parallel

and when m₁.m₂=-1  then both lines are perpendicular 

$-1.5x-3.5y=-31.5$

$-3.5y=1.5x-31.5$

$3.5y=-1.5x+31.5$

$y=-\frac{1.5}{3.5}x+\frac{31.5}{3.5}$

$y=-\frac{15}{35}x+\frac{315}{35}$

$y=-\frac{3}{7}x+9$

$so\space here \space m_1\ne m_2$

so both lines are not parallel

$now\space find \space its \space product\\ m_1 \cdot m_2= \frac{7}{3}\left(-\frac{3}{7}\right),\, {\color{Red} m_1 \cdot m_2= -1}$

therefore both lines are perpendicular 

The equation of the central street PQ is

${\color{Red} -1.5x-3.5y=-31.5} .$