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legs (sides other than hypotenuse) of right triangle are of lengths 16 cm & 8 cm. find the length of the side of largest square that be inscribed in the triangle



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See the diagram enclosed.  A square of largest size is possible only if you coincide two edges of triangle with the two edges of square.

Now area of triangle ABC = 1/2 *8 * 16  cm² = 64 cm²
Area of square = a²
area of triangle BDE =  1/2 (8-a) a = 4a - a²/2
area of triangle EFC = 1/2 (16 -a) a =  8a - a²/2
ΔABC = 64  = a² + 4a -a²/2 + 8a -a²/2  = 12 a 
a = 16/3
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