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Madhyamik Mathematics Suggestion – Circle-Related Theorems (Chapter 3) | WBBSE Class 10

15 Sept 2026 0 views

Madhyamik Mathematics Suggestion – Chapter 3 focuses on Circle-Related Theorems and includes important question-answer practice for WBBSE Class 10 Mathematics. This chapter covers fill-in-the-blanks, short-answer questions and descriptive problems based on circles, chords, tangents, cyclic quadrilaterals and related geometrical properties.

The questions below have been reorganized and rewritten in a clearer format so that students can use them more comfortably during revision.

Circle-Related Theorems – Chapter 3

বৃত্ত সম্পর্কিত উপপাদ্য | Madhyamik Mathematics Suggestion

Chapter 3 Mathematics
1 Mark Fill in the Blanks
5 Marks Descriptive Problems

Circle-Related Theorems: Important Areas

For this chapter, students should pay particular attention to the properties of tangents, chords, cyclic quadrilaterals, circles touching one another, angles related to the centre and circumference, and distances involving chords and tangents.

Topic Important Focus
Tangents Properties of tangents drawn from an external point
Chords Length of a chord and its distance from the centre
Cyclic Quadrilateral Relationships between opposite angles
Touching Circles Distance between centres and radii
Inscribed Polygon Properties of regular polygons inside a circle
Right-Angled Triangle Relation between circumcentre, radius and diameter

Fill in the Blanks – 1 Mark Questions

The following questions test important definitions and standard properties from the chapter.

1. Two tangents drawn to a circle from the same external point are ______ to each other.
Answer: Equal
2. The radius drawn to the point of contact of a tangent is ______ to the tangent.
Answer: Perpendicular
3. In a cyclic quadrilateral, the ______ angles are supplementary.
Answer: Opposite
4. Each side of a regular hexagon inscribed in a circle is equal to the circle's ______.
Answer: Radius
5. A straight line cannot intersect a circle at more than ______ points.
Answer: Two

Short-Answer Questions – 2 Marks

1. AB = 32 cm is a diameter of a circle. If AC = 32 cm and BC is a tangent, find BC.
Answer: Use the right-angle relationship between the radius and tangent together with the given geometrical construction to determine BC.
2. O is the circumcentre of triangle ABC. If ∠BAC = 50°, find ∠OBC.
Answer: Since O is the circumcentre, OA = OB = OC. Apply the isosceles-triangle angle relationships together with the central-angle and inscribed-angle theorem to obtain ∠OBC.
3. Two circles touch externally. The distance between their centres is 7 cm. If the radius of one circle is 4 cm, find the radius of the other circle.
Answer: For externally touching circles, the distance between the centres equals the sum of the two radii. Therefore, the second radius is 3 cm.

Long-Answer Questions – 5 Marks

These problems require proper geometrical reasoning, suitable theorems and step-by-step calculations. Students should practise writing the complete solution rather than only the final answer.

1. Distance Between Centres of Two Touching Circles

Two circles have radii 10 cm and 5 cm respectively. Find the distance between their centres when they touch internally and when they touch externally.
Solution:
For external contact, the distance between the centres is the sum of the radii:
AB = 10 + 5 = 15 cm
For internal contact, the distance between the centres is the difference between the radii:
AB = 10 − 5 = 5 cm
Therefore: External contact = 15 cm; Internal contact = 5 cm.

2. Three Externally Touching Circles

Three circles with centres A, B and C touch each other externally. If AB = 5 cm, BC = 7 cm and CA = 4 cm, determine the radii of the three circles.
Solution:
Let the radii of the circles with centres A, B and C be r₁, r₂ and r₃ respectively.
r₁ + r₂ = 5
r₂ + r₃ = 7
r₃ + r₁ = 4
Adding the three equations:
2(r₁ + r₂ + r₃) = 16
Therefore:
r₁ + r₂ + r₃ = 8
Hence:
r₁ = 1 cm,   r₂ = 4 cm,   r₃ = 3 cm

3. Finding the Length of a Chord

A circle has radius 10 cm. The perpendicular distance from the centre to a chord is 6 cm. Find the length of the chord.
Solution:
The perpendicular drawn from the centre of a circle to a chord bisects the chord. Therefore, if half the chord is x:
x² + 6² = 10²
x² = 100 − 36 = 64
x = 8 cm
Hence, the complete chord is:
2 × 8 = 16 cm
Answer: 16 cm

4. Distance of a Chord from the Centre

A circle has two chords of lengths 6 cm and 4 cm. If the distance of the shorter chord from the centre is 4 cm, find the distance of the longer chord from the centre.
Solution:
For a chord of length 4 cm, half the chord is 2 cm. If the radius is represented by r:
r² = 4² + 2² = 20
For the 6 cm chord, half the chord is 3 cm. If its distance from the centre is d:
d² + 3² = 20
d² = 11
Therefore:
d = √11 cm

5. Intersecting Chords and Diameter

Prove that two intersecting chords of a circle cannot bisect each other unless both chords are diameters.
Answer: Use the perpendicular-distance and chord properties of a circle. A chord passing through the centre is a diameter. When two chords bisect each other at their common point, the geometrical conditions force the common point to coincide with the centre. Hence both chords pass through the centre and therefore both are diameters.

6. Angles of a Cyclic Quadrilateral

Three consecutive angles of a cyclic quadrilateral are in the ratio 1 : 2 : 3. Find the ratio between the second angle and the fourth angle.
Solution:
Let the three consecutive angles be x, 2x and 3x. In a cyclic quadrilateral, opposite angles are supplementary. Therefore the first and fourth angles add up to 180°.

Since the second and fourth angles are also opposite angles, their sum is 180°.

Using the given ratio and the cyclic-quadrilateral property, determine the fourth angle and then express the required ratio.

7. Length of a Tangent

A point is situated 5 cm from the centre of a circle whose radius is 3 cm. Find the length of the tangent drawn from that point to the circle.
Solution:
Let the tangent length be t. The radius to the point of contact is perpendicular to the tangent.
t² + 3² = 5²
t² = 25 − 9 = 16
t = 4 cm
Answer: 4 cm

8. Circumcentre of a Right-Angled Triangle

In right-angled triangle ABC, ∠C = 90°. If O is the circumcentre and OA = 5 cm, find AB.
Solution:
In a right-angled triangle, the circumcentre lies at the midpoint of the hypotenuse. Therefore the circumradius is half the hypotenuse.
OA = AB ÷ 2
AB = 2 × 5 = 10 cm
Answer: 10 cm

9. Cyclic Quadrilateral and Central Angle

PQRS is a cyclic quadrilateral. QR is extended to T. If ∠QOS = 120°, find the required exterior angle ∠SRT.
Answer: Use the relationship between a central angle and the corresponding angle at the circumference, followed by the exterior-angle property of the cyclic quadrilateral.

10. Length of a Chord

A chord of a circle with radius 5 cm is 3 cm away from the centre. Find the length of the chord.
Solution:
Let half the chord be x. Then:
x² + 3² = 5²
x² = 25 − 9 = 16
x = 4 cm
Therefore:
Chord = 2 × 4 = 8 cm
Answer: 8 cm

11. Cyclic Quadrilateral Angle Problem

ABCD is a cyclic quadrilateral with centre O. If ∠BAC = 110° and ∠COD = 60°, find ∠BCD.
Answer: This problem should be solved by identifying the relevant arcs and applying the central-angle and inscribed-angle relationships carefully. Draw the figure first and mark the given angles before beginning the calculation.

Important Circle Theorems for Revision

Before attempting the longer problems, revise the following standard relationships from the chapter.

Property Key Relationship
Two tangents from an external point Their lengths are equal
Radius and tangent Radius at the point of contact is perpendicular to the tangent
Cyclic quadrilateral Opposite angles are supplementary
Chord and centre Perpendicular from the centre bisects a chord
Externally touching circles Distance between centres = sum of radii
Internally touching circles Distance between centres = difference of radii
Right triangle and circumcircle Hypotenuse is the diameter of the circumcircle

Formula-Based Practice

Tangent Length: PT² = OP² − OT²
Chord: (Half Chord)² = Radius² − Distance²
External Touching: Distance = r₁ + r₂
Internal Touching: Distance = |r₁ − r₂|
Revision Tip: While solving circle problems, always draw a neat diagram first. Mark the centre, radius, chord, tangent and given angles clearly. Many problems become easier once the correct theorem is identified from the diagram.

How to Practise Chapter 3 Effectively

Step 1: Revise the Theorems

Start by revising the basic properties of tangents, chords, cyclic quadrilaterals and circumcircles. Do not begin with lengthy numerical problems before the theorem statements are clear.

Step 2: Practise the Short Questions

Work through the 1-mark and 2-mark questions first. These questions help you identify which theorem is required for a particular situation.

Step 3: Solve the 5-Mark Problems

For descriptive questions, write the construction or reasoning step by step. Keep the mathematical statements clear and show the calculation leading to the final result.

Step 4: Recheck the Diagram

In geometry, an incorrectly interpreted diagram can lead to an incorrect theorem. Before calculating, identify the centre, radius, chord, tangent and relevant angles.

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Chapter 3 Quick Revision

Chapter 3 is centred on the geometry of circles. For effective revision, remember the standard properties of tangents, chords, cyclic quadrilaterals, touching circles and circumcircles. Practise both theorem-based questions and numerical applications so that you can recognize the correct method quickly.

Focus especially on diagrams and step-by-step reasoning when preparing the long-answer questions.

Frequently Asked Questions