# 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?

##### 1 Answer

For the third side to be the shortest, we require

#### Explanation:

The longest side of a right triangle is always the hypotenuse. So we know the length of the hypotenuse is

Let the unknown side length be

or

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:

Further, for third side to be smallest,

or

Combining all of these restrictions, we can deduce that in order for the third side to be the shortest, we must have