Is this geometry or what?

Geometry Level 3

A triangle has three integer side lengths and integer area If one of its sides is 21 long and the perimeter is 48, what is the length of its shortest side?


The answer is 10.

Ayush G Rai by Ayush G Rai , 20, India

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1 solution

Let the sides be 21 , x , 27 − x 21,x,27-x . The semi perimeter is 24. Hence, using Heron's formula, the area of the triangle is given by Area = 24 × 3 × ( 24 − x ) × ( x − 3 ) = 6 − 2 x 2 + 54 x − 144 = 6 2 ( x − 3 ) ( 24 − x ) \mbox{Area}\\=\sqrt{24\times3\times(24-x)\times(x-3)}\\=6\sqrt{-2x^2+54x-144}\\=6\sqrt{2(x-3)(24-x)}

This means that 2 ( x − 3 ) ( 24 − x ) 2(x-3)(24-x) is a perfect square. Trying all the values of x x from 4 to 23, we find that 10 10 is the length of the shortest side.

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good one...+1

Ayush G Rai - 4 years, 11 months ago

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