To find the area of the original rectangle, we need to determine its initial length and breadth. We are given information about how the rectangle transforms into a square and the area lost during this process.

Rectangle Dimensions and Transformation

Let the original length of the rectangle be \(L\) meters and the original breadth be \(B\) meters.

The area of the original rectangle is \(A_{\text{original}} = L \times B\) square meters.

• The length is reduced by 10 m, so the new length is \((L - 10)\) m.
• The breadth is reduced by 5 m, so the new breadth is \((B - 5)\) m.

After these reductions, the rectangle becomes a square. This means that the new length and the new breadth must be equal:

\[L - 10 = B - 5\]

We can rearrange this equation to express \(L\) in terms of \(B\):

\[L = B - 5 + 10\]

\[L = B + 5 \quad \text{(Equation 1)}\]

Area Loss Calculation

The area of the new square is \(A_{\text{new}} = (L - 10)(B - 5)\) square meters.

We are told that the rectangle loses 650 m\(^2\) of area during this process. This means the difference between the original area and the new area is 650 m\(^2\):

\[A_{\text{original}} - A_{\text{new}} = 650\]

Substitute the expressions for the areas:

\[L \times B - (L - 10)(B - 5) = 650\]

Now, expand the term \((L - 10)(B - 5)\):

\[(L - 10)(B - 5) = LB - 5L - 10B + 50\]

Substitute this back into the area loss equation:

\[LB - (LB - 5L - 10B + 50) = 650\]

Distribute the negative sign:

\[LB - LB + 5L + 10B - 50 = 650\]

The \(LB\) terms cancel out:

\[5L + 10B - 50 = 650\]

Add 50 to both sides:

\[5L + 10B = 650 + 50\]

\[5L + 10B = 700\]

Divide the entire equation by 5 to simplify:

\[L + 2B = 140 \quad \text{(Equation 2)}\]

Solving for Original Dimensions

Now we have a system of two linear equations with two variables:

1. \(L = B + 5\)
2. \(L + 2B = 140\)

Substitute the expression for \(L\) from Equation 1 into Equation 2:

\[(B + 5) + 2B = 140\]

Combine like terms:

\[3B + 5 = 140\]

Subtract 5 from both sides:

\[3B = 140 - 5\]

\[3B = 135\]

Divide by 3 to find the value of \(B\):

\[B = \frac{135}{3}\]

\[B = 45 \text{ meters}\]

Now that we have the breadth, substitute \(B = 45\) back into Equation 1 to find the length:

\[L = B + 5\]

\[L = 45 + 5\]

\[L = 50 \text{ meters}\]

Original Rectangle Area

Finally, we can calculate the area of the original rectangle using its length and breadth:

\[\text{Original Area} = L \times B\]

\[\text{Original Area} = 50 \text{ m} \times 45 \text{ m}\]

\[\text{Original Area} = 2250 \text{ m}^2\]

The area of the original rectangle is 2250 square meters.

This confirms our calculations align with the problem statement.