Question 697336
The area of a triangle is equal to (1/2)*base*height. For a right triangle, the base and the height are the legs.

Since this is an isosceles triangle, both legs are the same length. Let L = the length of a leg:

{{{(1/2)*L*L = 4}}} - divide both sides by 1/2:
{{{L^2 = 8}}} - take the square root of both sides:
{{{L = sqrt(8) = 2*sqrt(2)}}}.

The legs are both {{{2*sqrt(2)}}}. For the hypotenuse, you can use the Pythagorean theorem:

{{{a^2 + b^2 = c^2}}}, where {{{a = b = 2*sqrt(2)}}}:
{{{(2*sqrt(2))^2+(2*sqrt(2))^2 = c^2}}}
{{{8 + 8 = c^2}}}
{{{16 = c^2}}}, take the square root of both sides:
{{{4 = c}}}

The legs of the right triangle are {{{2*sqrt(2)}}}, and the hypotenuse is 4.

Follow-up edit: In this particular case, there are other ways to find the third side given the legs. Since this is a right isosceles (45-45-90) triangle, the ratios of the sides are going to be 1:1:{{{sqrt(2)}}}.

So, another way to find the hypotenuse is to multiply the length of a leg by {{{sqrt(2)}}}, which will also give 4 for the hypotenuse.

This specific method only works for right isosceles triangles, though. There's one other special case like this -- if you know the angles are 30, 60, and 90 degrees, the ratios of the sides are 1:{{{sqrt(3)}}}:2. The Pythagorean theorem works on any right triangle -- if you know any two sides you can always find the third side of a right triangle with it.