From a point P which is at a distance of 10 cm from the centre O of a circle of radius 6 cm, a pair of tangents PQ and PR to the circle at point Q and P respectively, are drawn. Then the area of the quadrilateral PQOR is equal to

Options

A. 30 sq.cm
B. 40 sq.cm
C. 24 sq.cm
D. 48 sq.cm

Answer

To find the area of quadrilateral PQOR, we can follow these steps:

### Step 1: Understand the Geometry
We have a circle with center O and radius 6 cm. Point P is outside the circle, 10 cm away from O. Tangents PQ and PR are drawn from point P to points Q and R on the circle.

### Step 2: Apply the Pythagorean Theorem
In triangle OPQ, we can apply the Pythagorean theorem since OP is the hypotenuse, and OQ is a radius (6 cm). The distance from P to the center O (OP) is given as 10 cm.

Using the Pythagorean theorem:
\[ OP^2 = OQ^2 + PQ^2 \]
Substituting the known values:
\[ 10^2 = 6^2 + PQ^2 \]
\[ 100 = 36 + PQ^2 \]
\[ PQ^2 = 100 - 36 = 64 \]
\[ PQ = \sqrt{64} = 8 \text{ cm} \]

### Step 3: Calculate the Area of Triangle OPQ
The area of triangle OPQ can be calculated using the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In triangle OPQ, we can take OQ (6 cm) as the base and PQ (8 cm) as the height:
\[ \text{Area}_{OPQ} = \frac{1}{2} \times OQ \times PQ = \frac{1}{2} \times 6 \times 8 = 24 \text{ cm}^2 \]

### Step 4: Calculate the Area of Quadrilateral PQOR
Since PQ and PR are equal (both are tangents from point P), triangle OPR will have the same area as triangle OPQ. Therefore, the area of quadrilateral PQOR is:
\[ \text{Area}_{PQOR} = \text{Area}_{OPQ} + \text{Area}_{OPR} = 24 + 24 = 48 \text{ cm}^2 \]

### Final Answer
The area of quadrilateral PQOR is **48 cm²**.

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