SOLUTION: Two vertices of an equilateral triangle are (3,4)and (-2,3) the third vertex can be

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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