## Problem Of The Week 32

**Four Equilateral triangles are formed taking one of their sides as the sides of the square, the third vertices of equilateral triangles being inside the square. The ratio of the area of fig formed by the third vertices of the triangles to that of the square is nearly**

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3/4 exactly. The triangles are formed in all the 3 dimensions. Now two of the vertices on either side of the square are at a distance (sqrt(3)/2*a+sqrt(3)/2*a)sqrt(3)a, ‘a’ being the side of the square. Now the altitude is sqrt(3)/2*a. So the area of triangle is 3/4*a^2 and the area of the square is a^2.

Celebrating LifeSeptember 29, 2008 at 8:24 am

@#Celebrating Life

acc to me

2-root(3):1 is the answer(almost 26.8%)

and the fig formed inside the square will be a

square itself(thru symmetry).

and the diagonal of the square is of length (root(3)-1)…

am i doing a mistake?

or u hav done a blunder?

wolverineOctober 1, 2008 at 8:26 am

yupp

the square formed will have the ratio of 0.27

outtimedOctober 1, 2008 at 8:51 am

yes ratio will be (2-3^.5)= 0.268

DufferOctober 12, 2008 at 11:54 am