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

Options

A. `65 cm^(2)`
B. `60 cm^(2)`
C. `30 cm^(2)`
D. `90 cm^(2)`

Answer

To find the area of quadrilateral PQOR, where P is a point outside the circle, O is the center of the circle, and PQ and PR are tangents drawn from point P to the circle, we can follow these steps:

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

### Step 2: Calculate the Length of the Tangents
Using the tangent-segment theorem, the length of the tangents from a point outside the circle to the points of tangency is given by the formula:
\[ PT = \sqrt{PO^2 - r^2} \]
where \( PO \) is the distance from point P to the center O, and \( r \) is the radius of the circle.

Here, \( PO = 13 \) cm and \( r = 5 \) cm.

Calculating the length of the tangent:
\[ PT = \sqrt{13^2 - 5^2} \]
\[ PT = \sqrt{169 - 25} \]
\[ PT = \sqrt{144} \]
\[ PT = 12 \text{ cm} \]

Thus, both PQ and PR are 12 cm long.

### Step 3: Calculate the Area of Triangle POR
To find the area of quadrilateral PQOR, we first need to find the area of triangle POR. The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

In triangle POR:
- Base OR = radius of the circle = 5 cm.
- Height = length of the tangent from P to the line OR, which is the length of the tangent PT = 12 cm.

Calculating the area of triangle POR:
\[ \text{Area}_{POR} = \frac{1}{2} \times 5 \times 12 \]
\[ \text{Area}_{POR} = \frac{1}{2} \times 60 \]
\[ \text{Area}_{POR} = 30 \text{ cm}^2 \]

### Step 4: Calculate the Area of Quadrilateral PQOR
The area of quadrilateral PQOR can be calculated as:
\[ \text{Area}_{PQOR} = \text{Area}_{POR} + \text{Area}_{\triangle PQO} + \text{Area}_{\triangle PRO} \]

Since triangles PQO and PRO are congruent (both have the same base and height), their areas are equal and can be calculated as:
\[ \text{Area}_{\triangle PQO} = \text{Area}_{\triangle PRO} = \frac{1}{2} \times 12 \times 5 = 30 \text{ cm}^2 \]

Thus,
\[ \text{Area}_{PQOR} = 30 + 30 + 30 = 90 \text{ cm}^2 \]

### Final Answer
The area of quadrilateral PQOR is **90 cm²**.

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