Let a=(4,2) b=(3,9) and c=(x,7) find x such that abc has a right angle at b

Plan:  to find the equation of the line that passes through the points (3,9) and (x,7).

 

This line will be perpendicular to the line that passes through the points (4,2) and (3,9).

 

First, find the slope of the line that passes through (4,2) and (3,9).

     slope  =  (9 - 2) / (3 - 4)  =  7 / -1  =  -7

 

Second, from this, find the slope of the line that passes through (3,9) and (x,7).

The slope of this line will be the negative reciprocal of the previous slope:  m  =  1/7.

 

Third, use the point-slope form to find the equation of the line that passes through

(3,9) and (x,7) using the point (3,9):  y - 9  =  (1/7)(x - 3)

 

Fourth, replace the y-value of this equation with the y-value of the point that we want to find.

   7 - 9  =  (1/7)(x - 3)

 

Last, solve this formula for x ...