**Given Information:**
- One side of the triangle measures 126m.
- The difference in lengths of the hypotenuse and the other side is 42m.

**To Find:**
- The measure of the two unknown sides of the triangle.
- The area of the triangle.

**Solution:**

Let's assume the two unknown sides of the triangle are a and b, and the hypotenuse is c.

**Using Pythagoras Theorem:**

According to Pythagoras theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

c² = a² + b²

Given that the difference in lengths of the hypotenuse and the other side is 42m, we can express this as:

c - b = 42

From these two equations, we can solve for a and b.

**Solving for a:**

We can rearrange the first equation to solve for a:

a² = c² - b²

Substituting the value of c - b from the second equation:

a² = (c - 42)² - b²

**Solving for b:**

Substituting the value of a² from the previous equation into the first equation:

(c - 42)² - b² + b² = 126²

Expanding and simplifying:

c² - 84c + 1764 = 126²

c² - 84c + 1764 - 126² = 0

Solving this quadratic equation will give us the value of c.

**Calculating the Area:**

Once we have the values of a, b, and c, we can calculate the area of the triangle using Heron's formula:

Area = √(s(s - a)(s - b)(s - c))

where s is the semi-perimeter of the triangle:

s = (a + b + c)/2

Substituting the values of a, b, and c into the formula will give us the area of the triangle.

**Verification:**

To verify the result using Heron's formula, we can calculate the area of the triangle using the given values of a, b, and c, and compare it with the calculated area. If both values match, our solution is correct.

Make sure to substitute the values correctly and double-check all calculations for accuracy.