Circles — Class 10 MCQs with Answers

Circles — important questions & MCQs with answers (Class 10 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   Number of tangents from an external point to a circle:

   A1B0C2DInfinite
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   Answer: 2

   Explanation: An external point lies outside the circle, so exactly two distinct tangent lines can be drawn to it — a point inside allows none, a point on the circle allows only one.

2. Q2 of 6Easy

   Tangent at any point of a circle is ___ to the radius at the point of contact.

   AParallelBPerpendicularCEqualDIntersecting
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   Answer: Perpendicular

   Explanation: The tangent-radius theorem: a tangent touches the circle at exactly one point, and at that point it is always perpendicular (⟂) to the radius — never parallel or merely equal in length.

3. Q3 of 6Medium

   In a quadrilateral ABCD circumscribing a circle, AB + CD equals:

   

   AAB onlyBAC + BDCBC + ADD2(BC + AD)
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   Answer: BC + AD

   Explanation: Tangent lengths from each vertex are equal, so every tangent segment is counted once on each side; summing opposite sides gives AB + CD = BC + AD — the Pitot theorem for tangential quadrilaterals.

4. Q4 of 6Medium

   Assertion (A): The tangents drawn at the two ends of a diameter of a circle are parallel. Reason (R): The tangent at any point of a circle is perpendicular to the radius through the point of contact.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true and R is the correct explanation of A

   Explanation: Both are true. Each tangent is perpendicular to the same diameter, so the two co-interior angles add to 90° + 90° = 180°, which makes the tangents parallel. R explains A.

5. Q5 of 6Hard

   PA and PB are tangents from P to a circle of radius 3 cm with centre O, and OP = 5 cm. The area of quadrilateral OAPB is:

   

   A6 cm²B24 cm²C12 cm²D15 cm²
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   Answer: 12 cm²

   Explanation: Each tangent is √(25 − 9) = 4 cm, and the figure is two right triangles of area 6 cm² each.

6. Q6 of 6Hard

   Assertion (A): Two concentric circles have radii 5 cm and 3 cm. The chord of the larger circle that touches the smaller circle is 6 cm long. Reason (R): In two concentric circles, the chord of the larger circle that touches the smaller circle is bisected at the point of contact.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: A is false but R is true

   Explanation: A is false: the radius 3 cm to the point of contact is perpendicular to the chord, so half the chord = √(5² − 3²) = 4 cm and the chord is 8 cm. R is true (Example 1).

Circles notes: key concepts (Class 10 Mathematics)

A tangent touches a circle at exactly one point, and everything in this chapter follows from the fact that it is perpendicular to the radius at that point. From it come the equal-tangent property and the standard results about tangents from an external point.

Tangent to a circle

A line meeting the circle at exactly one point, called the point of contact. It never enters the interior of the circle.

Secant

A line cutting the circle at two distinct points. As the two points approach each other the secant becomes a tangent.

Chord

A segment joining two points on the circle; the longest chord is the diameter, which passes through the centre.

Tangent perpendicular to radius

The tangent at any point is perpendicular to the radius drawn to that point — the theorem the whole chapter rests on.

Number of tangents from a point

None from inside the circle, exactly one from a point on it, and exactly two from any external point.

Equal tangents from an external point

The two tangent segments drawn from an external point to a circle are equal in length.

Tangent length

Found by Pythagoras in the right triangle formed by the radius, the tangent and the line to the centre.

Angle between the two tangents

The angle at the external point and the angle at the centre between the two radii are supplementary, adding to 180°.

Tangent to concentric circles

A chord of the larger circle that touches the smaller one is bisected at the point of contact.

Quadrilateral circumscribing a circle

In a quadrilateral whose four sides all touch a circle, the sums of the two pairs of opposite sides are equal.