The area of a right triangle is given by (1/2)bh, where b is the length of the base and h is the length of the height. In a right triangle, the legs can be considered the base and height. Using the Pythagorean Theorem, we can find the length of the other leg: a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse. Plugging in the values, we get 5^2 + b^2 = 13^2. This simplifies to 25 + b^2 = 169. Subtracting 25 from both sides, we get b^2 = 144. Taking the square root of both sides, we get b = 12. Now, the area of the triangle is (1/2)(5)(12) = 30.