SOLUTION: if two vertices of an equilateral triangle are (-4,-3) and(4,1), find the remaining vertex

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Question 1123419: if two vertices of an equilateral triangle are (-4,-3) and(4,1), find the remaining vertex




Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
There are two solutions (corresponding to points C and D) as shown in the drawing below.
Line AB has the equation +y+=+%281%2F2%29x+-+1+
Line CD has the equation +y+=+-2x+-+1+
+abs%28AB%29%5E2+=+%284-%28-4%29%29%5E2+%2B+%281-%28-3%29%29%5E2+=+64%2B16+=+80+
Using point C(Cx, Cy), the distance squared to C from B (and A) must be 80. Using B's coordinates:
+%28Cx-4%29%5E2+%2B+%28Cy-1%29%5E2+=+80+
++%28Cx%5E2-8Cx%2B16%29+%2B+%28Cy%5E2-2Cy%2B1%29+=+80+
Subs +Cy+=+-2Cx+-+1+ and Cy%5E2+=+4Cx%5E2+%2B+4Cx+%2B1 :

+%28Cx%5E2-8Cx%2B16%29+%2B+%284Cx%5E2%2B4Cx%2B1%29+%2B+%284Cx+%2B+2%29+%2B+1+=+80+
Which resolves to C at ( 2%2Asqrt%283%29+ , +%28-4%2Asqrt%283%29%29-1++ )

and D at ( -2%2Asqrt%283%29+ , +%284%2Asqrt%283%29%29-1++ )