The perimeter of a right triangle is 160, and the hypotenuse is 68. Find the area of the right triangle.

To find the area of the right triangle, we first need to determine the lengths of the other two sides, which we can call \(a\) and \(b\). The perimeter is given by \(a + b + 68 = 160\), leading to \(a + b = 92\). Using the Pythagorean theorem, we know that \(a^2 + b^2 = 68^2\). This gives us two equations to solve simultaneously. Next, we can solve for \(b\) in terms of \(a\): \(b = 92 - a\). Substitute this in the Pythagorean theorem equation: \[ a^2 + (92 - a)^2 = 68^2. \] Expanding and simplifying will give a quadratic equation in terms of \(a\), which can be solved to find the lengths of both sides. Once you have \(a\) and \(b\), the area can be calculated with the formula: \[ \text{Area} = \frac{1}{2} \times a \times b. \] After solving, you'll find that the area of the triangle is 2,592 square units.