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Circles – Class 10 Mathematics notes, important questions and MCQs

NCERT Class 10 Mathematics, chapter 10. Notes, key terms, important questions with answers and MCQs.

Chapter overview

This chapter studies tangents to a circle: a tangent touches the circle at exactly one point and is perpendicular to the radius at that point.

Tangent and secant

A secant cuts the circle at two points; a tangent touches it at one point.
From a point inside the circle, no tangent can be drawn; on the circle, one; outside, exactly two.
A circle can have infinitely many tangents.

Key theorems

The tangent at any point is perpendicular to the radius through the point of contact.
Lengths of the two tangents from an external point are equal.
The centre lies on the bisector of the angle between the two tangents.

Key terms

Tangent: a line meeting the circle at exactly one point
Point of contact: the point where a tangent touches the circle
Secant: a line meeting the circle at two points

Formulas

Tangent ⟂ radius at the point of contact
Tangent length from P: PT = √(OP² − r²)

Important questions with answers

A tangent PQ at P meets a line through the centre O at Q. If OP = 5 cm and OQ = 13 cm, find PQ.2 marks

The radius is perpendicular to the tangent, so PQ = √(13² − 5²) = √144 = 12 cm.

Prove that the lengths of tangents drawn from an external point to a circle are equal.3 marks

Let PA and PB be tangents from P to a circle with centre O. In right triangles OAP and OBP, OA = OB (radii), OP is common and ∠OAP = ∠OBP = 90°. By RHS congruence, △OAP ≅ △OBP, so PA = PB.

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.5 marks

Let XY be the tangent at P to a circle with centre O. Take any point Q on XY other than P. Q lies outside the circle (a tangent meets the circle only at P), so OQ > OP. Thus OP is the shortest distance from O to the line XY. The shortest segment from a point to a line is the perpendicular, so OP ⟂ XY.

Quick revision

Tangent ⟂ radius.
Two tangents from an external point are equal.
Outside point → 2 tangents; on the circle → 1; inside → 0.

Exam tips

Join the centre to the point of contact — a right angle appears immediately.
In a quadrilateral formed by two tangents and two radii, the angles at the centre and external point add to 180°.

Practice MCQs

How many tangents can a circle have?
1. 1
2. infinitely many
3. 4
4. 2
Show answer
Answer: (b) infinitely many. One at every point of the circle.
From a point inside a circle, the number of tangents is
1. 2
2. 1
3. 0
4. infinite
Show answer
Answer: (c) 0. No tangent can pass through an interior point.
The angle between a tangent and the radius at the point of contact is
1. 45°
2. 60°
3. 90°
4. 180°
Show answer
Answer: (c) 90°. Tangent is perpendicular to the radius.
From a point 13 cm from the centre, the tangent length to a circle of radius 5 cm is
1. 8 cm
2. 12 cm
3. 18 cm
4. 10 cm
Show answer
Answer: (b) 12 cm. √(169 − 25) = 12 cm.
Two tangents from P make 80° with each other. The angle between the radii at the points of contact is
1. 80°
2. 50°
3. 90°
4. 100°
Show answer
Answer: (d) 100°. 180° − 80° = 100°.
PA and PB are tangents from P with PA = 7 cm. Then PB =
1. 3.5 cm
2. cannot say
3. 14 cm
4. 7 cm
Show answer
Answer: (d) 7 cm. Tangents from an external point are equal.
A quadrilateral ABCD is drawn to circumscribe a circle. Then
1. AC = BD
2. AB + BC = CD + DA
3. AB = CD
4. AB + CD = BC + DA
Show answer
Answer: (d) AB + CD = BC + DA. Equal tangent segments give AB + CD = BC + DA.
Two concentric circles have radii 5 cm and 3 cm. The length of a chord of the larger circle touching the smaller one is
1. 8 cm
2. 4 cm
3. 6 cm
4. 10 cm
Show answer
Answer: (a) 8 cm. Half-chord = √(25 − 9) = 4, so chord = 8 cm.

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Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.