What is the length of the side opposite the right angle in a 45-45-90 triangle?

In a 45-45-90 triangle, which is an isosceles right triangle, the two legs are of equal length, and the angles measure 45 degrees, 45 degrees, and 90 degrees. The properties of this triangle allow us to derive the relationship between the length of the legs and the length of the hypotenuse.

In such triangles, if each leg has a length of ( n ), the hypotenuse can be calculated using the Pythagorean theorem. According to the theorem, the square of the length of the hypotenuse (( h )) is equal to the sum of the squares of the lengths of the other two legs. Therefore, we have:

[ h^2 = n^2 + n^2 = 2n^2 ]

Taking the square root of both sides gives us:

[ h = \sqrt{2n^2} = n\sqrt{2} ]

The hypotenuse is thus ( n\sqrt{2} ), which corresponds with the length of the side opposite the right angle in this specific triangle type.

Consequently, if we consider the length of the leg opposite the right angle in terms of the provided options, it is