P(2,1) , Q (4,-1) , R (3,2) are the vertices of a triangle and if through P and R lines parallel to opposite sides are drawn to intersect in S , then the area of PQRS , is

Options

A. 6
B. 4
C. 8
D. 12

Answer

To find the area of the quadrilateral PQRS formed by the triangle PQR and the lines drawn through points P and R parallel to the opposite sides, we can follow these steps:

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are given as:
- P(2, 1)
- Q(4, -1)
- R(3, 2)

### Step 2: Calculate the area of triangle PQR
We can use the formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates of points P, Q, and R:
\[ \text{Area} = \frac{1}{2} \left| 2(-1 - 2) + 4(2 - 1) + 3(1 + 1) \right| \]

### Step 3: Simplify the expression
Calculating each term:
- For the first term: \(2(-1 - 2) = 2 \times -3 = -6\)
- For the second term: \(4(2 - 1) = 4 \times 1 = 4\)
- For the third term: \(3(1 + 1) = 3 \times 2 = 6\)

Now, substituting back into the area formula:
\[ \text{Area} = \frac{1}{2} \left| -6 + 4 + 6 \right| = \frac{1}{2} \left| 4 \right| = \frac{4}{2} = 2 \]

### Step 4: Calculate the area of quadrilateral PQRS
Since PQRS is a parallelogram formed by extending lines from P and R parallel to the opposite sides, the area of PQRS is double the area of triangle PQR:
\[ \text{Area of PQRS} = 2 \times \text{Area of PQR} = 2 \times 2 = 4 \]

### Final Answer
The area of quadrilateral PQRS is \(4\) square units.

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