Madhyamik Mathematics Chapter 15 deals with the Theorems Related to Tangents to a Circle. This chapter contains important questions based on tangents, touching circles, chords, radii, centres and related geometrical properties.
This chapter-wise question-answer collection includes MCQs, True or False, Fill in the Blanks, Short Answer Questions and Long Answer Questions for WBBSE Class 10 Mathematics revision.
Theorems Related to Tangents to a Circle
The questions in this chapter focus on important geometrical relationships involving a circle and its tangent. Problems also include internally and externally touching circles, chord properties, angles and common tangents.
| Question Type | Marks | Chapter Focus |
|---|---|---|
| Multiple Choice Questions | 1 | Touching circles, tangents, chords and angles |
| True or False | 1 | Basic properties of tangents |
| Fill in the Blanks | 1 | Important circle and tangent properties |
| Short Answer Questions | 2 | Tangent length, chord and touching-circle problems |
| Long Answer Questions | 5 | Radius and parallel-chord geometry |
Multiple Choice Questions
MCQ Questions and Answers
1 Two circles touch each other externally. The radius of the smaller circle is 4 cm and the distance between the centres of the two circles is 2 cm. What will be the radius of the other circle?
2 Two circles touch each other internally. If the radii of the two circles are 7 cm and 4 cm respectively, what will be the distance between their centres?
3 From an external point P of a circle with centre O, two tangents PA and PB are drawn. If ∠APB = 60° and AP = 8 cm, what will be the length of AB?
4 In a circle with centre O, AB is a chord. A tangent PAQ is drawn at A. If ∠BAP = 48°, what is the value of ∠AOB?
5 In a circle with centre O, ABCD is a cyclic quadrilateral. A tangent PBQ is drawn at B. If ∠DBQ = 65°, what is the value of ∠BCD?
True or False
Important Statements
1 The two tangents drawn at the endpoints of a chord of a circle intersect each other.
2 The two tangents drawn at the endpoints of a diameter of a circle are parallel to each other.
3 The two tangents drawn from an external point to a circle have unequal lengths.
Fill in the Blanks
One-Mark Questions and Answers
1 If two circles touch each other internally, the point of contact and the centres of the two circles always lie along the same ________.
2 At any point of a circle, the tangent and the radius passing through the point of contact are inclined at a ________ angle.
3 The lengths of the two direct common tangents of two circles are ________.
4 If two circles neither intersect nor touch each other, there can be a maximum of ________ common tangents.
Short Answer Questions
2-Mark Questions with Solutions
1 A point is situated 13 cm away from the centre of a circle having radius 5 cm. Find the length of the tangent drawn from that point to the circle.
Radius of the circle, OA = 5 cm
Distance of the external point from the centre, OP = 13 cm
Let AP be the tangent drawn from the external point P to the circle.
Since the radius drawn to the point of contact is perpendicular to the tangent,
Therefore, by Pythagoras' theorem:
Therefore,
= 13² − 5²
= 169 − 25
= 144
Therefore,
2 In a circle with centre O, PQ is a chord and R is the midpoint of PQ. If OR = √11 cm and the radius is 2√3 cm, find the length of chord PQ.
OR = √11 cm
Radius, OQ = 2√3 cm
R is the midpoint of PQ
Since R is the midpoint of the chord PQ, the perpendicular drawn from the centre to the chord bisects the chord.
Therefore, in right-angled triangle ORQ:
= (2√3)² − (√11)²
= 12 − 11
= 1
Therefore,
Since R is the midpoint of PQ,
= 2 × 1
= 2 cm
3 Two circles touch each other internally. The radius of the larger circle is stated in the source question as 3 cm and the distance between the centres is 2 cm. Find the radius of the other circle.
Let the centre of the larger circle be A and its radius be:
Let the centre of the smaller circle be B and its radius be BC.
The distance between the two centres is:
Since the two circles touch each other internally,
= (6 − 2) cm
= 4 cm
Long Answer Questions
5-Mark Questions
1 In the given figure, O is the centre of the circle, OP ⟂ AB, AB = 6 cm and PC = 6 cm. Find the radius of the circle.
OP ⟂ AB
AB = 6 cm
PC = 6 cm
2 In a circle with centre O, two parallel chords AB and CD have lengths 10 cm and 24 cm respectively. The two chords are situated on opposite sides of the centre. If the distance between the chords AB and CD is 17 cm, determine the radius of the circle.
AB = 10 cm
CD = 24 cm
Distance between AB and CD = 17 cm
Important Concepts from Chapter 15
| Concept | Important Point |
|---|---|
| Externally Touching Circles | The question set includes problems involving the radii of externally touching circles and the distance between their centres. |
| Internally Touching Circles | Problems may require finding the smaller radius from the larger radius and the distance between the centres. |
| Tangent | A tangent and the radius drawn to the point of contact are perpendicular. |
| Two Tangents from an External Point | The two tangents drawn from the same external point have equal lengths. |
| Chord | A perpendicular from the centre to a chord bisects the chord. |
| Common Tangents | Two circles that neither intersect nor touch can have a maximum of four common tangents. |
Chapter 15 Quick Revision
For effective revision of this chapter, first understand the basic properties of tangents, radii, chords and touching circles. Then practise the objective questions before moving to numerical and figure-based problems.
Revise the relationship between a tangent and the radius.
Practise finding tangent length using the right triangle.
Practise problems involving the midpoint of a chord.
Revise internally and externally touching circle problems.
Remember the important properties of common tangents.
Carefully interpret the given geometry figure before solving.
Practice the Questions Systematically
Start with the 5 MCQs and 3 True or False questions to check your understanding of the basic concepts. Next, revise the four Fill in the Blanks questions because these cover important properties that can also help in solving numerical problems.
After that, practise the three 2-mark questions carefully. The first two problems require direct application of geometrical relationships, while the third deals with internally touching circles.
Finally, attempt the 5-mark questions involving the radius of a circle and parallel chords. These problems require careful interpretation of the given geometrical information.
Madhyamik Mathematics Suggestion Resources
Explore More Madhyamik Resources on Cademy
For more WBBSE Class 10 revision materials, students can explore the Madhyamik Suggestion 2027 – All Subjects resource on Cademy.
The resource includes subject-wise Madhyamik preparation areas, including Mathematics, along with the other major WBBSE Class 10 subjects.
Chapter 15 at a Glance
| Section | Questions Covered |
|---|---|
| MCQ | 5 questions with four options and answers |
| True or False | 3 statements with answers |
| Fill in the Blanks | 4 questions with answers |
| Short Answer | 3 questions, including complete solutions for the source-supported problems |
| Long Answer | 2 figure-based and geometry-based questions |