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By Heron\'s Formula, the area, A, of a triangle with sides a, b, c is given by A = square root[s(s − a)(s − b)(s − c)], where s = ½(a + b + c) is the semi-perimeter of the triangle.

Then s = ½(10 + 17 + 21) = 24, and A = 84.

now if you name the triangle PQR with PQ=10 QR=21 PR=17

and then draw a perpendicular from P to the side QR and name it PM

Now drop a perpendicular of length h onto the side of length 21.

We also have A = ½ × base × perpendicular height.

Hence A = 21h/2 = 84, from which h = 8.

Notice that the triangle above the square is similar to the whole triangle. (This follows because its base, the top of the square, is parallel to the base of the whole triangle.)

Let the square have side of length d.

Considering the ratio of altitude to base in each triangle, we have 8/21 = (8 − d)/d = 8/d − 1.

Therefore the length of the side of the square is 168/29=5.79=5.8