O A P B PA = PB OA ⊥ PA |PT|=√(d²−r²) AB+CD=BC+DA
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Chapter 10  ·  Class X Mathematics  ·  MCQ Practice

MCQ Practice Arena

Circles

Two Theorems, One Beautiful Chapter — Prove Fast, Score Full

📋 50 MCQs ⭐ 28 PYQs ⏱ 70 sec/Q
🚀 Start Practising 📖 Revise Notes First

MCQ Bank Snapshot

50Total MCQs
18Easy
20Medium
12Hard
28PYQs
70 secAvg Time/Q
7Topics
Easy 36% Medium 40% Hard 24%

Why Practise These MCQs?

CBSE Class XNTSEState BoardsOlympiad

Circles MCQs test both theorem recall and numerical application. The two cornerstone theorems (tangent ⊥ radius; tangents from external point are equal) appear in proof form (5 marks) AND MCQ form (1 mark) in the same CBSE paper. NTSE Geometry includes creative tangent-circle problems. Olympiad questions involve tangent constructions with elegant length relationships.

Topic-wise MCQ Breakdown

Tangent — Basic Concept & Secant6 Q
Tangent ⊥ Radius (Theorem & Application)10 Q
Tangents from External Point (Equal Length)12 Q
Length of Tangent Formula8 Q
Angle in Triangle with Tangent6 Q
Circumscribed Quadrilateral Property4 Q
Combined Circle + Triangle Problems4 Q

Must-Know Formulae Before You Start

Recall these cold before attempting MCQs — they appear in >70% of questions.

$\text{Tangent} \perp \text{radius at point of contact}$
$PA = PB\ (\text{tangents from external point P})$
$\text{Length of tangent} = \sqrt{d^2 - r^2}$
$\angle OPA = 90°\ (O = \text{centre, P = external, A = contact})$
$\text{Circumscribed quad: } AB+CD = BC+DA$

MCQ Solving Strategy

For every Circles MCQ, draw the circle with centre O, mark all tangent points, and write OA ⊥ PA immediately. For "find length of tangent" MCQs, use √(d²−r²) where d is distance from external point to centre. For "PA = PB" MCQs, identify the external point and both tangent lengths — they are always equal. For circumscribed quadrilateral MCQs: sum of opposite sides are equal.

⚠ Common Traps & Errors

Difficulty Ladder

Work through each rung in order — do not jump to Hard before mastering Easy.

① Easy

State theorems, verify PA = PB, identify tangent vs secant

② Medium

Find tangent length, angle at external point, OAP triangle

③ Hard

Circumscribed quadrilateral, combined triangle + circle, chord + tangent

★ PYQ

CBSE — proof + length numerical; NTSE — circle geometry reasoning

Continue Your Preparation

📖 Chapter Notes Revise concepts & theory 🧠 MCQ Practice Topic-wise Questions 📘✔️ Exercises Step-by-step NCERT solutions ✔️❌ True / False Concept-based True & False ➡ Next Chapter Areas Related to Circles
🎯 Knowledge Check

Maths — CIRCLES

100 Questions Class 10 MCQs
1
The tangent at any point of a circle is perpendicular to the
2
From an external point, the number of tangents that can be drawn to a circle is
3
The lengths of tangents drawn from an external point to a circle are
4
If the radius of a circle is \(5\) cm and length of tangent from external point is \(12\) cm, distance from centre to external point is
5
A line intersecting circle at exactly one point is
6
The line from centre to point of contact is
7
From a point on the circle, number of tangents is
8
If two tangents PA, PB from external point P, then angle OAP is
9
In \(\triangle OAP\), where O centre, A contact point, P external, OP\(^2 =\)
10
Tangents from external point to points of contact subtend equal angles at
11
If radius \(r = 13\) cm, tangent length \(l = 84\) cm, distance from centre is
12
The common external tangents to two circles are
13
Point inside circle can have
14
If PT = 10 cm, OT = radius = 24 cm, TO external point distance is
15
Angle between tangent and chord equals angle in
16
Two tangents from external point divide into equal
17
If distance from centre to external point 25 cm, radius 7 cm, tangent length
18
Number of tangents from external point is
19
The point of contact of tangent divides radius into
20
In figure with tangents PA, PB, then quadrilateral OAPB has angles at A,B
21
Length of tangent from P to circle centre O radius 5 cm, PO = 13 cm is
22
Tangent segments from external point are
23
A secant intersects circle at
24
If two circles touch externally, distance between centres equals
25
The normal at point of contact is
26
From point X, tangent 20 cm, distance to centre 25 cm, radius is
27
Angles subtended by tangents at external point are
28
Circle has number of tangents equal to
29
If tangent length 12 cm, radius 16 cm, distance OP is
30
Point of contact lies on
31
Two concentric circles have chords of one circle which are
32
Line through centre bisecting chord is
33
Perpendicular from centre to chord
34
Equal chords subtend equal angles at
35
Number of circles through 3 non-collinear points is
36
If PT tangent, OT radius, angle at T is
37
In two tangents PA, PB, triangles OAP, OBP are
38
Distance between points of contact A,B from external P is calculated by
39
If circles touch internally, centre distance
40
Chord perpendicular distance from centre relates to
41
Tangent at point P on circle, chord PQ, angle between tangent-chord equals angle in
42
Length of two tangents from P are each \(l\), radius \(r\), OP =
43
Infinite tangents possible because
44
If PO = 10 cm, r = 8 cm, tangent length
45
Quadrilateral formed by two radii and tangents has sum of opposite angles
46
Secant from external point intersects at two points, unlike tangent
47
In NCERT Circles chapter, main theorems are
48
Point lying on circle has tangents
49
If tangent touches at A, then OA extended is normal
50
Basic definition: circle is points equidistant from
51
The formula for the area of a circle of radius \(r\) is
52
If the radius of a circle is doubled, its area becomes
53
The value of \(\pi\) generally used in NCERT problems is
54
A sector of a circle is bounded by
55
The total angle at the centre of a circle is
56
The area of a sector of angle \(\theta^\circ\) and radius \(r\) is
57
A sector whose angle is less than \(180^\circ\) is called
58
A sector whose angle is more than \(180^\circ\) is called
59
A segment of a circle is bounded by
60
The smaller region formed by a chord and an arc is called
61
The area of a minor segment is found by
62
The area of a major segment is
63
Which of the following is a chord of a circle?
64
The longest chord of a circle is
65
If the diameter of a circle is \(14\) cm, its radius is
66
The area of a semicircle of radius \(r\) is
67
The unit of area related to circles is
68
If the radius of a circle is zero, its area is
69
A quadrant of a circle subtends an angle of
70
The area of a quadrant of radius \(r\) is
71
Which concept is most useful in finding shaded regions?
72
The boundary length of a circle is called
73
The formula for circumference of a circle is
74
The area of a sector is directly proportional to
75
A circle is divided into equal sectors. The areas of the sectors are
76
The region enclosed between two concentric circles is called
77
The area of a ring depends on
78
In exams, shaded region problems mostly involve
79
Which value of \(\pi\) is used when radius is a multiple of 7?
80
The area of a sector with angle \(180^\circ\) is
81
Which figure is formed by a diameter and an arc?
82
The area of a minor segment is always
83
To find the area of a major sector, we usually
84
Which mathematical idea links angles and areas in circles?
85
A wheel of radius \(r\) completes one full rotation. The area covered relates to
86
The region enclosed by a chord and the major arc is
87
Which topic is essential before studying this chapter?
88
Area related to circles belongs to which branch of mathematics?
89
If the radius is given in metres, the area will be in
90
Which of the following is NOT a part of this chapter?
91
The area of a circle increases when
92
A semicircle subtends an angle of
93
The formula \(\frac{\theta}{360}\times \pi r^2\) is based on
94
In shaded region problems, the first step should be
95
Which shape helps in finding segment area?
96
The area of a circle is zero when
97
Which figure gives maximum area for a fixed perimeter?
98
The formula for area of a ring is
99
Which problems are most common in board exams from this chapter?
100
This chapter mainly strengthens which skill?
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Class 10 Maths Ch 10 MCQs: Circles (With Answers & Test)
Class 10 Maths Ch 10 MCQs: Circles (With Answers & Test) — Complete Notes & Solutions · academia-aeternum.com
This set of 50 multiple choice questions is designed to give Class 10 students thorough practice of Chapter 10 “Circles” from the NCERT Mathematics textbook. The questions systematically cover all key ideas: basic definitions of a circle, radius and chord, properties of tangents, the number of tangents from a point, and the main theorems on tangents to a circle.​ Each MCQ is accompanied by a clear answer and a concise explanation so that learners can quickly identify mistakes and revise…
🎓 Class 10 📐 Mathematics 📖 NCERT ✅ Free Access 🏆 CBSE · JEE
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Exam tip: Sharing chapter notes with your study group creates a reinforcement loop. Teaching a concept is the fastest path to mastering it.

Frequently Asked Questions

A circle is the set of all points in a plane that are at a fixed distance from a fixed point called the centre.

The centre is the fixed point from which all points on the circle are equidistant.

The radius is the line segment joining the centre of the circle to any point on its circumference.

The diameter is a chord passing through the centre of the circle and is twice the radius.

A chord is a line segment joining any two points on the circumference of a circle.

The diameter is the longest chord of a circle.

A secant is a line that intersects a circle at two distinct points.

A tangent is a line that touches a circle at exactly one point.

The point where a tangent touches the circle is called the point of contact.

A tangent has exactly one common point with the circle.

A secant has exactly two common points with the circle.

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

The angle is always a right angle (90°).

Exactly one tangent can be drawn from a point on the circle.

No tangent can be drawn from a point inside the circle.

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    CIRCLES – Learning Resources

    Detailed Notes
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    📚
    ACADEMIA AETERNUM तमसो मा ज्योतिर्गमय · Est. 2025
    Sharing this chapter
    Class 10 Maths Ch 10 MCQs: Circles (With Answers & Test)
    Class 10 Maths Ch 10 MCQs: Circles (With Answers & Test) — Complete Notes & Solutions · academia-aeternum.com
    This set of 50 multiple choice questions is designed to give Class 10 students thorough practice of Chapter 10 “Circles” from the NCERT Mathematics textbook. The questions systematically cover all key ideas: basic definitions of a circle, radius and chord, properties of tangents, the number of tangents from a point, and the main theorems on tangents to a circle.​ Each MCQ is accompanied by a clear answer and a concise explanation so that learners can quickly identify mistakes and revise…
    🎓 Class 10 📐 Mathematics 📖 NCERT ✅ Free Access 🏆 CBSE · JEE
    Share on
    WhatsApp
    Telegram
    X / Twitter
    Facebook
    Instagram
    Threads
    Pinterest
    LinkedIn
    Quora
    Discord
    Reddit
    Email
    academia-aeternum.com/class-10/mathematics/circles/mcqs/ Copy link
    💡
    Exam tip: Sharing chapter notes with your study group creates a reinforcement loop. Teaching a concept is the fastest path to mastering it.

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