NCERT  ·  Mathematics  ·  Class X  ·  Ch.10

Circles
MCQ Series

Tangents · Number of Tangents · Chords & Radii · Angle Properties — 50 expert questions crafted for CBSE Boards and JEE preparation.

⚡ Start Quiz 📊 Analytics
50
Questions
40 min
Suggested Time
5
Topics Covered
34
Concept Check (Theorems)
Question Intelligence

Quiz Analytics

A data-driven breakdown of all 50 questions by difficulty, exam origin and topic distribution.

📈 Distribution Overview

50
Total Questions
Concept Check (Theorems)
34
Reasoning with Right Triangles
10
Mixed Geometry Insight
6

🗂 Topic Coverage

Tangent–Radius Perpendicularity
32%
Number & Length of Tangents
28%
Chords, Secants & Angles
16%
Touching Circles & Concentric
12%
Circle Basics & Definitions
12%
34
Concept Check (Theorems)
10
Reasoning with Right Triangles
6
Mixed Geometry Insight
Conceptual Framework

Key Concept Highlights

6 foundational pillars that power every question in this quiz. Understand these, and the answers follow naturally.

📐
Tangent and Radius
The tangent at any point of a circle is perpendicular to the radius drawn to the point of contact, so radius–tangent pairs always form right angles.
📏
Number & Equality of Tangents
From an external point exactly two tangents can be drawn to a circle, from a point on the circle only one tangent, and from a point inside the circle no tangent is possible; tangents from the same external point are equal in length.
📎
Right Triangle with Tangent
If OT is a radius and PT a tangent from an external point P, then triangle OPT is right-angled at T and satisfies OP^2 = OT^2 + PT^2, which is used to relate radius, tangent length and distance of the point from the centre.
🧵
Chords, Secants & Bisectors
A secant meets the circle at two points, a chord joins two points on the circle, and any line through the centre that is perpendicular to a chord bisects that chord and the angle subtended by equal chords at the centre are equal.
🎯
Tangent–Chord Angle Property
The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment of the circle formed by that chord.
⭕
Touching & Concentric Circles
If two circles touch externally, the distance between their centres equals the sum of their radii, while for internal touching it equals the difference of radii; concentric circles share the same centre but have different radii.
Pedagogical Value

Why MCQs Matter

Multiple-choice questions are not mere guessing games — they are the sharpest diagnostic tool available to a competitive exam aspirant.

~6–8%

of Class 10 Maths Geometry weightage under “Circles” in school exams, periodic tests and CBSE board papers.

Learning Outcomes

What You Will Learn

By completing this quiz set you will have exercised all the following competencies.

01 State and use the theorem that the tangent at any point of a circle is perpendicular to the radius through that point.
02 Determine the number of tangents from points lying inside, on and outside a circle and apply the fact that tangents from an external point are equal in length.
03 Solve right-triangle problems involving radius, tangent length and distance of an external point from the centre using the Pythagoras relation in triangle OPT.
04 Differentiate between chords, secants and tangents and apply results about perpendiculars from the centre bisecting chords and equal chords subtending equal angles.
05 Use distance relations for externally and internally touching circles and understand properties of concentric circles.
06 Apply the tangent–chord angle property and simple angle-chasing to quadrilaterals formed by radii and tangents, such as OAPB with two tangents from an external point.
07 Recall the basic definition of a circle as the set of all points in a plane equidistant from a fixed point called the centre and connect this with radius-based reasoning in problems.
Exam Preparation

Strategy & Preparation Tips

5 evidence-based strategies to maximise your score in CBSE Boards and JEE.

01
Step 01
Lock Core Tangent Theorems
Memorise the perpendicular-radius theorem and the equal-tangents-from-an-external-point theorem; almost all MCQs in this chapter reduce to these two ideas.
02
Step 02
Always Mark the Right Angle
In every figure with a tangent and radius, immediately mark the 90° at the point of contact so you can set up OP^2 = OT^2 + PT^2 without hesitation.
03
Step 03
Classify the Given Line First
Before answering, decide whether the line is a chord, secant or tangent; this single step usually tells you which theorem or property to apply.
04
Step 04
Use Simple Distance Checks
For touching or concentric circles, quickly compare centre distance with sum or difference of radii instead of overthinking the geometry.
05
Step 05
Sketch Mini Figures for MCQs
For statements about two tangents PA, PB and quadrilateral OAPB, draw a quick labelled sketch; symmetric right triangles reveal equal lengths and angle relations at a glance.

Ready to Test Your Mastery?

50 questions  ·  Elapsed timer  ·  Instant scored results

⚡ Begin Circles Quiz
🎯 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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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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