Multiple choice

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 points Q and R, respectively, are drawn. Then the area of the quadrilateral PQOR is equal to

  1. 30 sq. cm

  2. 40 sq. cm

  3. 24 sq. cm

  4. 48 sq. cm
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D Correct answer
Explanation

The quadrilateral PQOR is composed of two congruent right-angled triangles, PQR and POR, where the radius OQ is 6 cm and the hypotenuse OP is 10 cm. Using the Pythagorean theorem, PQ = sqrt(10^2 - 6^2) = 8 cm. The area of one triangle is 1/2 * base * height = 1/2 * 6 * 8 = 24 sq. cm, so the total area is 24 * 2 = 48 sq. cm.

AI explanation

The tangent PQ is perpendicular to the radius OQ, making triangle OQ a right triangle. By the Pythagorean theorem, the length of tangent PQ is the square root of (10 squared minus 6 squared), which equals the square root of 64 or 8 cm. The area of right triangle OQ is one half times base OQ times height PQ, giving one half times 6 times 8, which is 24 square cm. Since the quadrilateral PQOR consists of two identical right triangles, its total area is 2 times 24, yielding 48 sq. cm.

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