From a point P which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. Then the area of the quadrilateral PQOR is :

Options

A. 60 `cm^(2)`
B. 65 `cm^(2)`
C. 30 `cm^(2)`
D. 32.5 `cm^(2)`

Answer

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

### Step 1: Understand the Geometry
We have a circle with center O and radius 5 cm. Point P is outside the circle at a distance of 13 cm from O. Tangents PQ and PR are drawn from point P to the circle.

### Step 2: Draw the Diagram
Draw the circle with center O. Mark point P outside the circle such that OP = 13 cm. Draw the tangents PQ and PR from point P to the circle, touching the circle at points Q and R respectively.

### Step 3: Identify Key Segments
- OP (distance from center to point P) = 13 cm
- OQ (radius) = 5 cm
- OR (radius) = 5 cm

### Step 4: Use the Right Triangle Property
The radius drawn to the tangent point is perpendicular to the tangent. Therefore, triangles OPQ and OPR are right triangles.

### Step 5: Apply the Pythagorean Theorem
In triangle OPQ:
- OP is the hypotenuse (13 cm)
- OQ is one leg (5 cm)
- PQ is the other leg, which we need to find.

Using the Pythagorean theorem:
\[ OP^2 = OQ^2 + PQ^2 \]
\[ 13^2 = 5^2 + PQ^2 \]
\[ 169 = 25 + PQ^2 \]
\[ PQ^2 = 169 - 25 = 144 \]
\[ PQ = \sqrt{144} = 12 \text{ cm} \]

### Step 6: Calculate the Area of Triangle OPQ
The area of triangle OPQ can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, base OQ = 5 cm and height PQ = 12 cm.
\[ \text{Area}_{OPQ} = \frac{1}{2} \times 5 \times 12 = \frac{60}{2} = 30 \text{ cm}^2 \]

### Step 7: Area of Triangle OPR
Since triangles OPQ and OPR are congruent (by the properties of tangents and the fact that they share the side OP), the area of triangle OPR is also 30 cm².

### Step 8: Calculate the Area of Quadrilateral PQOR
The area of quadrilateral PQOR is the sum of the areas of triangles OPQ and OPR:
\[ \text{Area}_{PQOR} = \text{Area}_{OPQ} + \text{Area}_{OPR} \]
\[ \text{Area}_{PQOR} = 30 + 30 = 60 \text{ cm}^2 \]

### Final Answer
The area of quadrilateral PQOR is **60 cm²**.

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