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Question 1116921: The perpendicular bisector of the interval CD has equation 4x-3y+16=0. C has coordinates (-9,10). find the coordinates of D
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! A PICTURE IS USUALLY HELPFUL:
The line with equation  <-->  has slope  .
It is easy to see that it passes through point(-4,0).
I can plot points of that line by adding  to the x-coordinate
(moving 3 spaces to the right),
and adding  to the y-coordinate
(moving 4 spaces up) from any point on the line, including point (-4,0).
That allows me to graph that line, along with point  .
As the line with equation <--> and slope .
as it is the perpendicular bisector of segment CD,
it is perpendicular to the line CD containing segment CD, and points C and D.
Perpendicular lines have slopes whose product is  ,
so the slope  of the line containing points C and D is such that
 -->  .
That slope would allow me to plot points of line CD (and graph line CD),
by adding to the x-coordinate
(moving 4 spaces to the right),
and adding  to the y-coordinate
(moving 3 spaces down) from any point on the line, including point C(-9,10).
The points added and line CD are shown in green below.
 .
From point to point on each line, I moved
units in one direction and perpendicular to that direction,
for a distance (along the line) of  ,
so I see that point C is at a distance  from the blue perpendicular bisector line,
so the point at distance on the other side of that line is point  .
PROBABLY "FORMULAS" AND EQUATIONS ARE EXPECTED TO BE SHOWN:
The line with equation <--> and slope .
As it is the perpendicular bisector of segment CD,
it is perpendicular to the line CD containing segment CD, and points C and D.
Perpendicular lines have slopes whose product is ,
so the slope of the line containing points C and D is such that
--> .
Knowing the coordinates of and the slope of the line,
we can write the equation of line CD in point-slope form as
 or  .
Multiplying both sides of the equal sign times , and rearranging, we get
 and  .
The intersection of linr CD and the perpendicular bisector of segment CD
is the midpoint of CD, and is given by
 .
Adding up the first equation times 3 plus the second times 4, we get
 -->  ,
and substituting that into , we get
 -->  -->  .
So, the midpoint of CD,  has  and  .
The coordinates of the midpoint of a segment CD, with  and  are given by
 and  .
Substituting the known coordinates of C and M,
 -->  --> 
and
 -->  -->  .
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