### Theory:

A rectangle can be constructed using two set of measurements.

**The two instances are**:

**1.**When the length and breadth is known.

**2.**When a side and a diagonal is known.

Method \(1\): When the length and breadth is known

Step \(1\): Draw a rough diagram using the measurements known.

Step \(2\): Draw a line segment \(AB\) of length \(10\) \(cm\).

Step \(3\): With \(A\) as centre, draw a perpendicular line.

Step \(4\): With \(A\) as centre and with \(4\) \(cm\) as radius, draw an arc on the perpendicular line and mark the intersection as \(D\).

Step \(5\): With \(D\) as centre and with \(10\) \(cm\) as radius, draw an arc. Similarly, with \(B\) as centre and with \(4\) \(cm\) as radius, cut the existing arc and mark the intersection as \(C\).

Step \(6\): Now join \(CD\) and \(BC\) to obtain the desired rectangle.

**To find the area of the rectangle**:

\(\text{Area of the rectangle} = \text{Length} \times \text{Breadth}\)

\(\text{Area of the rectangle} = l \times b\)

We know that, \(l = 10\) \(cm\), and \(b = 4\) \(cm\).

Now, \(\text{Area of the rectangle} = 10 \times 4\)

\(= 40\) \(cm^2\)

Method \(2\): When a side and a diagonal is known

Let us construct rectangle with a side as \(8\) \(cm\) and the diagonal as \(10\) \(cm\). Let us also calculate the area of the obtained rectangle.

Step \(1\): Draw a rough diagram with the known measurements.

Step \(2\): Draw a line segment \(AB\) of length \(8\) \(cm\).

Step \(3\): With \(A\) as centre and with \(10\) \(cm\) as radius, draw an arc.

Step \(4\): With \(B\) as centre, draw a perpendicular line until it meets the arc already drawn. Mark the intersection as \(C\).

Step \(5\): Measure \(BC\). In this case, \(BC = 6\) \(cm\), With \(A\) as centre and with \(6\) \(cm\) as radius, draw an arc. Similarly, with \(C\) as centre and with \(8\) \(cm\) as radius, cut the arc. Mark the intersection as \(D\).

Step \(6\): Join \(AD\) and \(CD\) to form the rectangle.

**To find the area of the rectangle**:

\(\text{Area of the rectangle} = \text{Length} \times \text{Breadth}\)

\(\text{Area of the rectangle} = l \times b\)

We know that, \(l = 8\) \(cm\), and \(b = 6\) \(cm\).

Now, \(\text{Area of the rectangle} = 8 \times 6\)

\(= 48\) \(cm^2\)