Point `P(p ,0),Q(q ,0),R(0, p),S(0,q)` from.

Options

A. parallelogram
B. rhombus
C. cyclic quadrilateral
D. none of these

Explanation

To determine the type of quadrilateral formed by the points \( P(p, 0) \), \( Q(q, 0) \), \( R(0, p) \), and \( S(0, q) \), we can follow these steps:

### Step 1: Plot the Points
We start by plotting the points on the coordinate plane:
- Point \( P(p, 0) \) lies on the x-axis.
- Point \( Q(q, 0) \) also lies on the x-axis.
- Point \( R(0, p) \) lies on the y-axis.
- Point \( S(0, q) \) also lies on the y-axis.

### Step 2: Connect the Points
Next, we connect the points in the order \( P \to Q \to S \to R \to P \) to form a quadrilateral.

### Step 3: Analyze the Angles
Now, we analyze the angles formed by these points:
- The angle at point \( P \) between lines \( PQ \) and \( PS \) is \( 90^\circ \) because \( PQ \) is horizontal and \( PS \) is vertical.
- Similarly, the angle at point \( Q \) between lines \( QR \) and \( QS \) is also \( 90^\circ \).
- The angle at point \( R \) between lines \( RS \) and \( RP \) is \( 90^\circ \).
- The angle at point \( S \) between lines \( SP \) and \( SQ \) is also \( 90^\circ \).

### Step 4: Check Opposite Angles
Since all angles are \( 90^\circ \), we can conclude that:
- The sum of opposite angles \( \angle P + \angle R = 90^\circ + 90^\circ = 180^\circ \).
- The sum of opposite angles \( \angle Q + \angle S = 90^\circ + 90^\circ = 180^\circ \).

### Step 5: Conclusion
Since the opposite angles of the quadrilateral formed by points \( P, Q, R, S \) add up to \( 180^\circ \), we can conclude that the quadrilateral is a cyclic quadrilateral.

### Final Answer
The quadrilateral formed by the points \( P(p, 0) \), \( Q(q, 0) \), \( R(0, p) \), and \( S(0, q) \) is a **cyclic quadrilateral**.

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