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Question 938100: Two vertices of an equilateral triangle are (3,4)and (-2,3) the third vertex can be
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! We know one side of the triangle, connecting vertices  and  .
The length of side AB is
Side AB is a segment of a line with slope
 .
The midpoint of AB, point P, has coordinates
 and  .
The equilateral triangles with side AB are ABD and ABC,
with C and D on opposite sides of segment AB.
Visualizing the location of vertices C and D is easy,
but I could not think of a simple way to calculate their coordinates.
The perpendicular bisector of AB contains the altitude of those triangles.
That perpendicular bisector is a line perpendicular to AB,
and passing through point P (the midpoint of AB).
Being perpendicular to a line with slope  ,
that altitude is part of a line with slope  .
 .
Since angle PAC is a  ,
 -->
So the length of PC and PD is  ,
but what are the coordinates of C and D?
 Since the slope of CD is  ,  ,
and in those two green right triangles
 ---> ---> --->
Then,
 (rounded),
 (rounded),
 (rounded), and
 (rounded).
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