If ABCD be a rectangle and P, Q, R, S be the mid-points of AB, BC, CD and DA respectively, then the area of the quadrilateral PQRS is equal to

Options

A. `1/3` are (ABCD)
B. `3/4` area (ABCD)
C. `1/2` area (ABCD)
D. area (ABCD)

Answer

To find the area of the quadrilateral PQRS formed by the midpoints of the sides of rectangle ABCD, we can follow these steps:

### Step 1: Define the dimensions of the rectangle
Let the length of rectangle ABCD be \(2x\) and the breadth be \(2y\).

### Step 2: Identify the midpoints
The midpoints of the sides are:
- \(P\) is the midpoint of \(AB\)
- \(Q\) is the midpoint of \(BC\)
- \(R\) is the midpoint of \(CD\)
- \(S\) is the midpoint of \(DA\)

From the dimensions:
- \(AP = PB = \frac{2x}{2} = x\)
- \(BQ = QC = \frac{2y}{2} = y\)
- \(CR = RD = x\)
- \(DS = SA = y\)

### Step 3: Calculate the area of rectangle ABCD
The area of rectangle ABCD is given by:
\[ \text{Area}_{ABCD} = \text{Length} \times \text{Breadth} = 2x \times 2y = 4xy \]

### Step 4: Calculate the area of triangles formed
The quadrilateral PQRS is formed by the midpoints, and we can see that it divides the rectangle into four triangles:
1. Triangle \(APS\)
2. Triangle \(PBQ\)
3. Triangle \(QCR\)
4. Triangle \(RDS\)

The area of each triangle can be calculated as:
\[ \text{Area of each triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times x \times y \]
Since there are four such triangles, the total area of the triangles is:
\[ \text{Total area of triangles} = 4 \times \left(\frac{1}{2} \times x \times y\right) = 2xy \]

### Step 5: Calculate the area of quadrilateral PQRS
The area of quadrilateral PQRS is the area of rectangle ABCD minus the area of the four triangles:
\[ \text{Area}_{PQRS} = \text{Area}_{ABCD} - \text{Total area of triangles} = 4xy - 2xy = 2xy \]

### Conclusion
Thus, the area of quadrilateral PQRS is:
\[ \text{Area}_{PQRS} = 2xy \]