Let's solve this step-by-step.

Let:

• $l$ = original length of the rectangle
• $b$ = original breadth of the rectangle

Step 1: Use the perimeter formula

The perimeter of a rectangle is given by: $2(l + b) = 44$ Dividing both sides by 2: $l + b = 22$

Step 2: Use the area increase information

When the length is increased by 4 cm and the breadth is increased by 2 cm, the new length and breadth become $l + 4$ and $b + 2$, respectively. The area increase is given as 72 cm².

The original area of the rectangle is: $A = l \times b$ The new area is: $A_{\text{new}} = (l + 4)(b + 2)$

The increase in area is: $A_{\text{new}} - A = 72$

Expanding $A_{\text{new}}$: $(l + 4)(b + 2) = l \times b + 72$ $lb + 2l + 4b + 8 = lb + 72$ Canceling $lb$ from both sides: $2l + 4b + 8 = 72$ Subtracting 8 from both sides: $2l + 4b = 64$ Dividing both sides by 2: $l + 2b = 32$

Step 3: Solve the equations

Now we have a system of equations:

1. $l + b = 22$
2. $l + 2b = 32$

Subtract the first equation from the second: $(l + 2b) - (l + b) = 32 - 22$ $b = 10$

Now substitute $b = 10$ into the first equation: $l + 10 = 22$ $l = 12$

Conclusion

The original length of the rectangle is 12 cm.

Would you like further explanation or details on any part of this solution?

Here are some related questions for further practice:

1. How would the solution change if the perimeter was different?
2. What if the increase in area was a different amount, like 50 cm²?
3. How do you find the dimensions if only the area and one dimension are given?
4. What if only the perimeter and area are known? How would you approach finding length and breadth?
5. How does changing both dimensions affect the perimeter?

Tip: When solving systems of equations, isolating one variable and substituting is a common and effective strategy.