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 (let's call them 'a' and 'b'). Knowing that the perimeter is the sum of all sides (a + b + 68 = 160), we can simplify this to a + b = 92. Next, we also use the Pythagorean theorem: \(a^2 + b^2 = 68^2 = 4624\). Solving for 'a' and 'b' together with these two equations will yield the dimensions. From \(a + b = 92\), we express 'b' as \(b = 92 - a\). Substituting into the Pythagorean theorem: \(a^2 + (92 - a)^2 = 4624\), we expand and combine like terms: \[ a^2 + (8464 - 184a + a^2) = 4624 \] \[ 2a^2 - 184a + 3840 = 0 \] Dividing everything by 2 simplifies to \(a^2 - 92a + 1920 = 0\). Using the quadratic formula (\(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)), we find the values for 'a', and subsequently 'b'. Once we have 'a' and 'b', we can determine the area of the triangle using the formula \(\text{Area} = \frac{1}{2}ab\). The calculated area will provide the value for the right triangle! Can you figure out the rest? It’s a fun puzzle!