ABCD is a `//` gram. P and Q are two points on AB and CD such that AP : PB = 2:3 , DQ = QC = 4 : 1 Find the ratio of area of Δ DPQ and '// ABCD

Answer

To find the ratio of the area of triangle DPQ to the area of parallelogram ABCD, we will follow these steps:

### Step 1: Understand the Ratios
Given:
- \( AP : PB = 2 : 3 \)
- \( DQ : QC = 4 : 1 \)

This means that if we let \( AP = 2x \) and \( PB = 3x \), then \( AB = AP + PB = 2x + 3x = 5x \).

Similarly, for \( DQ \) and \( QC \), if we let \( DQ = 4y \) and \( QC = 1y \), then \( CD = DQ + QC = 4y + 1y = 5y \).

### Step 2: Area of Parallelogram ABCD
The area of the parallelogram ABCD can be calculated using the base and height. Since \( AB \) and \( CD \) are equal, we can use the length \( AB \) as the base.

Let the height from point D to line AB be \( h \). Thus, the area of parallelogram ABCD is:
\[ \text{Area}_{ABCD} = \text{Base} \times \text{Height} = AB \times h = 5x \times h \]

### Step 3: Area of Triangle DPQ
To find the area of triangle DPQ, we can use the formula for the area of a triangle:
\[ \text{Area}_{DPQ} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Here, the base can be taken as \( PQ \) and the height will be the perpendicular distance from point D to line PQ.

Since \( DQ : QC = 4 : 1 \), we can say that \( PQ \) is divided in the same ratio. The length of \( PQ \) can be calculated as follows:
- The total length of \( CD \) is \( 5y \).
- Since \( DQ = 4y \) and \( QC = 1y \), the length of \( PQ \) can be considered as \( PQ = DQ + AP \) where \( AP = \frac{2}{5} \times AB \).

### Step 4: Calculate the Ratio of Areas
The area of triangle DPQ can be expressed in terms of the area of the parallelogram:
\[ \text{Area}_{DPQ} = \frac{1}{2} \times PQ \times h' \]
Where \( h' \) is the height from D to line PQ.

Since the triangles formed by the segments are similar, the ratio of the areas of triangle DPQ to parallelogram ABCD can be simplified using the ratios of their bases:
\[ \frac{\text{Area}_{DPQ}}{\text{Area}_{ABCD}} = \frac{4}{5} \]

### Final Ratio
Thus, the ratio of the area of triangle DPQ to the area of parallelogram ABCD is:
\[ \text{Ratio} = \frac{4}{5} \]