Question

The largest side of a right triangle is `a^2+b^2` and other side is `2ab`. What condition will make the third side to be the smallest side?

Answer 1

For the third side to be the shortest, we require `(1+sqrt2)|b|>absa>absb` (and that `a` and `b` have the same sign).

Explanation:

The longest side of a right triangle is always the hypotenuse. So we know the length of the hypotenuse is `a^2+b^2.`

Let the unknown side length be `c.` Then from the Pythagorean theorem, we know

`(2ab)^2+c^2=(a^2+b^2)^2`
or
`c=sqrt((a^2+b^2)^2-(2ab)^2)`
`color(white)c=sqrt(a^4+2a^2b^2+b^4-4a^2b^2)`
`color(white)c=sqrt(a^4-2a^2b^2+b^4)`
`color(white)c=sqrt((a^2-b^2)^2)`
`color(white)c=a^2-b^2`

We also require that all side lengths be positive, so

• `a^2+b^2>0`
  `=>a!=0 or b!=0`
• `2ab>0`
  `=>a,b>0 or a,b<0`
• `c=a^2-b^2>0`
  `<=>a^2>b^2`
  `<=>absa>absb`

Now, for any triangle, the longest side must be shorter than the sum of the other two sides. So we have:

`color(white)(=>)2ab+"       "c color(white)(XX)>a^2+b^2`
`=>2ab+(a^2-b^2)>a^2+b^2`
`=>2ab color(white)(XXXXXX)>2b^2`

`=>{(a>b"," if b > 0), ( a < b"," if b < 0 ):}`

Further, for third side to be smallest, `a^2-b^2 < 2ab`
or `a^2-2ab+b^2 < 2b^2` or `a-b < sqrt2b` or `a < b(1+sqrt2)`

Combining all of these restrictions, we can deduce that in order for the third side to be the shortest, we must have `(1+sqrt2)|b|>absa>absb and (a,b<0 or a,b>0).`