Tangent Theorems of a Circle is an important part of Madhyamik Mathematics Chapter 15. This chapter focuses on tangents, circles, chords, radii and the geometric relationships formed when a tangent touches a circle.
For WBBSE Class 10 Mathematics preparation, students should be comfortable with the basic properties of tangents as well as numerical problems involving the radius, distance between centres and lengths of chords and tangents.
Tangent Theorems of a Circle – Basic Concept
A tangent is a straight line that touches a circle at a particular point. The point where the tangent touches the circle is called the point of contact.
Several important results in this chapter are based on this relationship. Questions may also involve two tangents drawn from an external point, touching circles internally or externally, common tangents and chords.
Important Properties of Tangents
| Concept | Important Result |
|---|---|
| Tangent and radius | The radius through the point of contact is perpendicular to the tangent. |
| Two tangents from an external point | The two tangent segments have equal lengths. |
| Two circles touching internally | The centres and the point of contact lie on the same straight line. |
| Two non-intersecting circles | They can have a maximum of four common tangents. |
MCQ – Multiple Choice Questions
1. External Contact Between Two Circles
Two circles touch each other externally. The radius of the smaller circle is 4 cm and the distance between their centres is 10 cm. Find the radius of the other circle.
```2. Internal Contact Between Two Circles
Two circles touch each other internally. Their radii are 7 cm and 4 cm. Find the distance between their centres.
```Distance between centres = Difference of radii
= 7 − 4
= 3 cm
3. Two Tangents from an External Point
From an external point P, two tangents PA and PB are drawn to a circle with centre O. If ∠APB = 60° and AP = 8 cm, find the length of AB.
```4. Tangent and Chord
In a circle with centre O, AB is a chord. A tangent PAQ is drawn at A. If ∠BAP = 48°, find the value of ∠AOB.
```5. Tangent to a Cyclic Quadrilateral
ABCD is a cyclic quadrilateral in a circle with centre O. A tangent PBQ is drawn at B. If ∠DBQ = 65°, find ∠BCD.
```True or False
1. The tangents drawn at the endpoints of a chord of a circle intersect each other.
2. The 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
1. If two circles touch internally, the point of contact and the centres of the two circles are always ________.
2. At the point of contact, the tangent and the radius 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, they can have a maximum of ________ common tangents.
Short Answer Questions
1. Length of a Tangent from an External Point
```A point is situated 13 cm from the centre of a circle whose radius is 5 cm. Find the length of the tangent drawn from the point to the circle.
PA² = OP² − OA²
= 13² − 5²
= 169 − 25
= 144
Therefore, PA = 12 cm.
2. Length of a Chord
```In a circle with centre O, PQ is a chord and R is the midpoint of PQ. If OR = 2√3 cm and the radius of the circle is 2√3 cm, find the length of PQ.
OR ⟂ PQ.
In right triangle ORP:
OP² = OR² + PR²
(2√3)² = (2√3)² + PR²
12 = 12 + PR²
Therefore PR = 0.
3. Radius of an Internally Touching Circle
```Two circles touch each other internally. The larger circle has radius 3 cm and the distance between their centres is 2 cm. Find the radius of the other circle.
Distance between centres = Difference between the radii.
Therefore, if the larger radius is taken exactly as written in the question, R = 3 cm and centre distance = 2 cm:
r = R − 2 = 3 − 2 = 1 cm.
Long Answer Questions
1. Finding the Radius of a Circle
```In the given figure, O is the centre of the circle, OP is perpendicular to AB, AB = 6 cm and PC = 6 cm. Find the radius of the circle.
2. Radius of a Circle from Two Parallel Chords
```In a circle with centre O, two parallel chords AB and CD have lengths 10 cm and 24 cm respectively. The chords lie on opposite sides of the centre, and the distance between the two chords is 17 cm. Find the radius of the circle.
Solution
The perpendicular from the centre of a circle to a chord bisects the chord. Therefore, half of the 10 cm chord is 5 cm and half of the 24 cm chord is 12 cm.
For the 10 cm chord:
r² = x² + 5²
For the 24 cm chord:
r² = y² + 12²
Since the chords are on opposite sides of the centre:
x + y = 17
Thus the problem can be solved by combining the two right-triangle relationships with the given 17 cm separation.
r² − x² = 25
r² − y² = 144
Subtracting:
y² − x² = 119
(y − x)(y + x) = 119
Since y + x = 17:
17(y − x) = 119
y − x = 7
Therefore:
y = 12 and x = 5.
r² = x² + 5²
= 5² + 5²
= 50
Therefore,
r = 5√2 cm.
Quick Revision Chart
Key Formulas for Chapter 15
Chapter 15 Exam Preparation
For Madhyamik Mathematics Chapter 15, students should practise the relationship between a tangent and radius, equal tangents from an external point, internally and externally touching circles, common tangents and chord-based numerical problems.
While solving geometry questions, always draw or carefully study the given figure first. Mark the radius, tangent, chord and centre clearly before applying a theorem. This makes the calculation easier and helps avoid mistakes in angle and length problems.
For wider WBBSE Class 10 preparation, you can also use Cademy's Madhyamik Suggestion 2027 All Subjects resource, which includes Mathematics among the subject-wise preparation resources.
Revision Focus: Remember the core relationships first. Once the tangent-radius theorem, equal tangent theorem, chord property and circle-contact relationships are clear, many Chapter 15 numerical problems become much easier to handle.