Question

Let G denote the centroid of triangle ABC. If triangle ABG is equilateral with side length 2, then determine the perimeter of triangle ABC.

Answer 1

We know from Euclidean Geometry and the properties of a centroid that GC=2GM. Now GM=sin60*GA=
`√(3)`. Hence CM=GC+GM=3*
`√(3) /`. Now, since GM is normal to AB, we have by the pythagoeran theorem that:

`AC^2=AM^2+GM^2=BM^2+GM^2=BC^2`
Hence, we calculate from this that AC^2= 28, hence AC=2*
`√(7)`=BC. Thus, the perimeter of the triangle is 2+4*
`√(7)`.