Circles — Questions & Answers

Sample questions from Circles

1. Q1. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. The length of PQ is:

   • A.7 cm
   • B.√119 cm✓
   • C.119 cm
   • D.13 cm
   Solution

   The tangent is perpendicular to the radius at the point of contact, so triangle OPQ is right-angled at P with OQ as the hypotenuse. PQ = √(12² − 5²) = √119 cm. Option D treats 5 and 12 as the two legs and gives 13, which would be the answer only if the right angle were at O.

2. Q2. Tangents PA and PB are drawn from an external point P to a circle of radius 6 cm with centre O. If ∠APB = 60°, then OP equals:

   • A.6√2 cm
   • B.6√3 cm
   • C.12 cm✓
   • D.18 cm
   Solution

   OP bisects ∠APB, so ∠APO = 30° and sin 30° = OA/OP = 6/OP, giving OP = 12 cm.

3. Q3. Why is a diameter the longest chord?

   • A.It passes through the centre and has maximum possible distance across the circle✓
   • B.It is a tangent
   • C.It has zero length
   • D.It is perpendicular to all chords
   Solution

   A chord is longest when its distance from centre is zero.

4. Q4. A circle is inscribed in a triangle ABC right angled at B, with AB = 8 cm and BC = 6 cm. The radius of the circle is:

   • A.4 cm
   • B.3 cm
   • C.2 cm✓
   • D.1 cm
   Solution

   AC = 10 cm, and for a right triangle r = (AB + BC − AC)/2 = (8 + 6 − 10)/2 = 2 cm.

5. Q5. A tangent from P to a circle has length 12 cm. The other tangent from P is: Verify the related circle calculation:

   • A.12 cm✓
   • B.6 cm
   • C.24 cm
   • D.cannot be found
   Solution

   Tangents from an external point are equal.

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