To solve this problem, we need to first set up the equation using the Pythagorean theorem. This tells us that in a right-angled triangle, the square of the length of the hypotenuse — the side opposite the right angle — equals the sum of the squares of the lengths of the other two sides. In equation form, this is:
a² + b² = c²
In this case, we don't know the lengths of the two shorter sides, which we'll call a and b. We do know that the length of the longer of these two sides exceeds the shorter by 10cm, so we can express the length of the longer side (b) in terms of the shorter side (a) as follows:
b = a + 10
We also know that the length of the hypotenuse (c) is 50cm.
Now we can substitute the expression for b into the Pythagorean theorem, giving us:
a² + (a + 10)² = 50²
Solving this quadratic equation using the method of your choice, you find that the length of the shorter side (a) is 30cm. Substituting this back in gives you the length of the longer side (b), which is 40cm.
Finally, to find the area of the triangle, we can use the formula:
1/2 * base * height
Taking the shorter side as the base and the longer side as the height, we have:
1/2 * 30cm * 40cm = 600 square cm.
So, the shorter side of the triangle is 30cm, the longer side is 40cm, and its area is 600 square cm.