`PQRS `is a rectangle. Points A, B, C and D are the mid points of sides `PQ, QR, RS and PS `respectively. If area of `DeltaPQR ` is `48 cm^2`, then what is the area (in cm) of `DeltaBCD`?

Options

A. 24
B. 6
C. 16
D. 12

Answer

To find the area of triangle BCD in rectangle PQRS, we can follow these steps:

### Step 1: Understand the Geometry
We have a rectangle PQRS, where points A, B, C, and D are the midpoints of sides PQ, QR, RS, and PS, respectively.

### Step 2: Find the Area of Rectangle PQRS
Given that the area of triangle PQR is 48 cm², we know that triangle PQR is half of the rectangle PQRS because the diagonal PR divides the rectangle into two equal triangles.

\[ \text{Area of rectangle PQRS} = 2 \times \text{Area of triangle PQR} = 2 \times 48 \text{ cm}^2 = 96 \text{ cm}^2 \]

### Step 3: Determine the Area of Quadrilateral DBRS
Since A, B, C, and D are midpoints, quadrilateral DBRS is half of the rectangle PQRS. Therefore, the area of quadrilateral DBRS is:

\[ \text{Area of DBRS} = \frac{1}{2} \times \text{Area of rectangle PQRS} = \frac{1}{2} \times 96 \text{ cm}^2 = 48 \text{ cm}^2 \]

### Step 4: Relate Triangle BCD to Quadrilateral DBRS
Triangle BCD is formed within quadrilateral DBRS. Since BD is parallel to RS and both triangles share the same base BD and height from point C to line RS, we can use the property of triangles in parallel lines.

### Step 5: Calculate the Area of Triangle BCD
The area of triangle BCD is half of the area of quadrilateral DBRS:

\[ \text{Area of triangle BCD} = \frac{1}{2} \times \text{Area of DBRS} = \frac{1}{2} \times 48 \text{ cm}^2 = 24 \text{ cm}^2 \]

### Final Answer
The area of triangle BCD is **24 cm²**.

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