What is the equation of the line that is perpendicular to the line passing through #(-5,3)# and #(4,9)# at midpoint of the two points?

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Answer 1 · 2016-04-07T15:59:21

#y=-1 1/2x+2 1/4#

The slope a line that is perpendicular to a given line would be the inverse slope of the given line

#m = a/b# the perpendicular slope would be #m =-b/a#

The formula for the slope of a line based upon two coordinate points is

#m = (y_2-y_1)/(x_2-x_1)#

For the coordinate points #(-5,3) and (4,9)#
#x_1 = -5#
#x_2 = 4#
#y_1 = 3#
#y_2 = 9#

#m = (9-3)/(4-(-5))#

#m = 6/9#

The slope is #m = 6/9#
the perpendicular slope would be the reciprocal (-1/m)
#m = -9/6#

To find the midpoint of the line we must use the midpoint formula

#((x_1+x_2)/2,(y_1+y_2)/2)#

#((-5+4)/2,(3+9)/2)#

#(-1/2,12/2)#

#(-1/2,6)#

To determine the equation of the line use the point slope form
#(y-y_1)=m(x-x_1)#

Plug in the midpoint in order to find the new equation.
#(-1/2,6)#

#(y-6)=-9/6(x-(-1/2))#

#y-6=-9/6x-9/12#

#ycancel(-6)cancel(+6)=-1 1/2x-3/4 +3#

#y=-1 1/2x+2 1/4#