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

To solve for the area of the right triangle, we first use the given perimeter and hypotenuse. Let the other two sides be and . The perimeter is given by:

Thus,

Next, we can use the Pythagorean theorem:

Now we have a system of equations:

We can express in terms of from the first equation:

Now, we substitute for in the second equation:

Expanding the second term:

This simplifies to:

Now, subtract 4624 from both sides:

Dividing the entire equation by 2 simplifies to:

Next, we can use the quadratic formula where :

Calculating the discriminant:

Now, we compute :

Calculating both possible values:

So, either and or and .

Finally, the area of a right triangle can be calculated using:

Thus, the area of the right triangle is .