The sum of the lengths of the three sides of a right-angled triangle is 16 cm. The sum of the squaresof the lengths of the three sides of the triangle is 98 cm2.What is the area, in cm2, of the triangle?

Let's denote the sides of the triangle as a, b, and c, where c is the hypotenuse.

From the problem, we have two equations:

1. a + b + c = 16 (since the sum of the lengths of the three sides is 16 cm)

2. a² + b² + c² = 98 (since the sum of the squares of the lengths of the three sides is 98 cm²)

We also know from the Pythagorean theorem that in a right-angled triangle, a² + b² = c².

We can substitute c² from the Pythagorean theorem into the second equation to get 2c² = 98, which simplifies to c² = 49, so c = 7 cm.

Substitute c = 7 into the first equation to get a + b = 16 - 7 = 9 cm.

The area of a right-angled triangle is given by 1/2 * base * height. Here, the base and height are a and b, and we know that a * b = area * 2.

We also know from the Pythagorean theorem that a * b = (c² - (a² + b²) / 2) = (49 - 9² / 2) = 20.5 cm².

So, the area of the triangle is 20.5 cm².