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
a. 65 cm²
b. 60 cm²
c. 30 cm²
d. 42 cm²

Solution:

Given, P is a point at a distance of 13 cm from the centre O of a circle of radius 5 cm.

PQ and PR are the tangents of the circle.

We have to find the area of the quadrilateral.

From the figure,

OQ = OR = radius.

We know that the radius of a circle is perpendicular to the tangent at the point of contact.

So, ∠PQO = 90°

∠PRO = 90°

Given, PO = 13 cm

Also, OP = OQ = 5 cm

In triangle PQO,

PQO is a right triangle with Q at right angle.

By pythagorean theorem,

(hypotenuse)² = (base)² + (height)²

(13)² = PQ² + (5)²

169 = PQ² + 25

PQ² = 169 - 25

PQ² = 144

Taking square root,

PQ = 12 cm

We know that the tangents through an external point to a circle are equal.

i.e., PQ = PR

So, PR = 12 cm

From the figure,

Area of quadrilateral PQRS = area of △POQ + area of △POR

In general, area of triangle = 1/2 × base × height

Area of △POQ = 1/2 × 5 × 12 = 60/2 = 30 cm²

Area of △POR = 1/2 × 5 × 12 = 60/2 = 30 cm²

Area of quadrilateral = 30 + 30 = 60 cm²

Therefore, the area of quadrilateral PQRS is 60 cm²

✦ Try This: In a circle with radius 13 cm, two equal chords are at a distance of 5 cm from the centre. Find the lengths of chords.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10

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NCERT Exemplar Class 10 Maths Exercise 9.1 Problem 4

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 a. 65 cm², b. 60 cm², c. 30 cm², d. 42 cm²

Summary:

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 60 cm²

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