Question 93150
Well, on this problem you do have to make at least one assumption:
The given side of 3 meters is (or is not) the hypotenuse.
If you assume that it is the hypotenuse, it is therefore the longest of the three sides of the right triangle.
Since the other two sides are consective integers and each is less then 3, they can only be 1 and 2 mters in length.
Well, as soon as you apply the Pythagorean theorem, you soon see that this cannot be so, because {{{3^2}}} does not equal {{{1^2 + 2^2}}}.
So the 3-meter side is one of the legs, right?
Now you have a choice: Is it the shorter leg or the longer leg (not the hypotenuse)?
Let's try the shorter leg.
So the hypotenuse, being the longest side must be n+1 and the other leg must be n. Applying the Pythagorean theorem: {{{c^2 = a^2+b^2}}}
{{{(n+1)^2 = n^2 + 3^2}}}
{{{n^2+2n+1 = n^2 + 9}}} Subtract {{{n^2}}} from both sides.
{{{2n+1 = 9}}}
{{{2n = 8}}}
{{{n = 4}}} and...
{{{n+1 = 5}}}
So, the hypotenuse is 5 meters,and the other leg is 4 meters.

If you were familiar with Pythagorean triplets, you could easily have guessed this result.