Assertion (A): A tangent to a circle intersects the circle at exactly one point. Reason (R): The point where a tangent touches the circle is called the point of contact.

A and R both correct, R does not explain A

Assertion (A): The lengths of tangents drawn from an external point to a circle are equal. Reason (R): The radius drawn to the point of contact is perpendicular to the tangent.

A and R both correct, R explains A

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This quiz for Chapter 10, 'Circles', contains 40 unique questions in game mode, covering various difficulty levels from easy to hard. There are also solved NCERT book questions.

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The quiz covers all major topics from the 'Circles' chapter, including definitions of secants and tangents, the property of radius being perpendicular to the tangent, and the equality of tangent lengths from an external point.

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Home › CBSE › Class 10 › Maths › CIRCLES

CBSE · Class 10 · 🧮 Maths · Chapter 10

CIRCLES

Non-intersecting lineSecant of a circleTangent to a circlePoint of contactRadius perpendicular to tangentLengths of tangents from external point

Chapter 10, 'Circles', introduces fundamental concepts related to circles and lines in a plane. Students learn about non-intersecting lines, secants, and tangents. Key theorems covered include the property that the tangent at any point of a circle is perpendicular to the radius through the point of contact, and that the lengths of tangents drawn from an external point to a circle are equal. This chapter forms a crucial foundation for higher geometry.

Introduction: Circle aur Line ke beech ke Relations

Jab ek circle aur ek line ek plane mein hote hain, toh unke beech teen possibilities ho sakti hain:

Non-intersecting Line:
Line aur circle ka koi common point nahi hota.
Ye line circle ko touch ya cut nahi karti.
Example: Ek train track aur uske bagal mein rakha hua coin (agar track coin ko touch na kare).

Secant:
Line circle ko do distinct points par intersect karti hai.
Ye line circle ke andar se guzar jaati hai.
Example: Pizza ko cut karne wali knife, jo do points par crust ko cut karti hai.

Tangent:
Line circle ko exactly ek point par touch karti hai.
Is common point ko point of contact kehte hain.
Tangent circle ko cut nahi karti, sirf touch karti hai.
Example: Cycle ka wheel jab road ko touch karta hai, toh road ek tangent ki tarah act karti hai.

Key Differences:

| Feature | Non-intersecting Line | Secant | Tangent | |---------------------|-----------------------|-----------------------|-----------------------| | Common Points | 0 | 2 | 1 | | Relation with Circle| No contact | Intersects | Touches | | Example | Train track & coin | Pizza knife | Cycle wheel & road |

Tangent, secant ka special case hai jahan chord ke dono end points coincide kar jaate hain.

📖Definition

Point of Contact: Wo common point jahan tangent line circle ko touch karti hai.

💡Tip

CBSE mein aksar tangent, secant, aur chord ke definitions aur unke beech ke differences par 1-mark ke questions aate hain. Clear hona chahiye!

Tangent to a Circle: Definition aur Properties

Tangent ek aisi line hoti hai jo circle ko sirf ek point par touch karti hai. Ye point 'point of contact' kehlata hai.

Theorem 10.1: Radius is Perpendicular to Tangent

Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Matlab: Agar ek circle hai jiska center O hai aur ek tangent XY hai jo circle ko point P par touch karti hai, toh radius OP, tangent XY par perpendicular hoga. Yaani, \(\angle OPX = \angle OPY = 90^\circ\).
Significance: Ye theorem circles aur tangents se related most fundamental aur frequently used property hai. Iske bina koi bhi tangent problem solve karna mushkil hai.

Proof of Theorem 10.1:

Given: Ek circle jiska center O hai, aur ek tangent XY jo circle ko point P par touch karti hai.
To Prove: \(OP \perp XY\).
Construction: Tangent XY par point P ke alawa koi aur point Q lo. O aur Q ko join karo.
Steps:

Point Q circle ke bahar hoga. Agar Q circle ke andar hota, toh XY circle ko do points par cut karti, jo tangent ki definition ke against hai.
Iska matlab hai ki \(OQ > OP\) (kyunki OP radius hai aur OQ circle ke bahar tak ja raha hai).
Ye baat har point Q ke liye true hai jo XY par hai (P ke alawa).
Toh, OP, point O se line XY tak ka shortest distance hai.
Geometry mein, shortest distance hamesha perpendicular hota hai.
Hence, \(OP \perp XY\).

Properties derived from Theorem 10.1:

Point of contact par ek hi tangent draw ki ja sakti hai.
Radius aur tangent ke beech ka angle hamesha \(90^\circ\) hota hai.
Agar ek line radius ke end point par perpendicular hai, toh wo line circle ki tangent hogi.

❗Important

Theorem 10.1 ka proof CBSE exams mein aksar 2-3 marks mein pucha jaata hai. Isko acche se practice kar lena.

🚧Misconception

Students aksar confuse ho jaate hain ki kaunsi line perpendicular hai. Hamesha yaad rakho, radius point of contact par tangent ke perpendicular hota hai, na ki tangent radius ke.

Number of Tangents from a Point on a Circle

Ek point se circle par kitne tangents draw kiye ja sakte hain, ye us point ki position par depend karta hai:

Point circle ke andar hai (Inside the circle):

Agar point P circle ke andar hai, toh us point se koi tangent draw nahi ki ja sakti.
Kyunki agar koi line us point se pass hogi aur circle ko touch karegi, toh wo circle ko do points par cut karegi (secant ban jayegi).

Point circle par hai (On the circle):

Agar point P circle par hai, toh us point se exactly ek tangent draw ki ja sakti hai.
Ye tangent point P par radius ke perpendicular hogi (Theorem 10.1).

Point circle ke bahar hai (Outside the circle / External point):

Agar point P circle ke bahar hai, toh us point se exactly do tangents draw ki ja sakti hain.
Ye dono tangents circle ko do different points par touch karengi.

Theorem 10.2: Lengths of Tangents from an External Point

Statement: The lengths of tangents drawn from an external point to a circle are equal.
Matlab: Agar ek external point P se circle par do tangents draw ki jaati hain, say PQ aur PR, jahan Q aur R points of contact hain, toh \(PQ = PR\).
Significance: Ye theorem bhi bahut important hai aur is par based questions aksar exams mein aate hain.

Proof of Theorem 10.2:

Given: Ek circle jiska center O hai. Ek external point P se do tangents PQ aur PR draw ki gayi hain, jahan Q aur R points of contact hain.
To Prove: \(PQ = PR\).
Construction: O ko P, Q, aur R se join karo.
Steps:

\(OQ \perp PQ\) aur \(OR \perp PR\) (By Theorem 10.1, radius is perpendicular to tangent at point of contact).
Consider \(\triangle OQP\) aur \(\triangle ORP\).
\(\angle OQP = \angle ORP = 90^\circ\) (Right angles).
\(OQ = OR\) (Radii of the same circle).
\(OP = OP\) (Common side).
Toh, \(\triangle OQP \cong \triangle ORP\) (By RHS congruence criterion).
Since triangles congruent hain, unke corresponding parts equal honge.
Hence, \(PQ = PR\) (By CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

Important Inferences from Theorem 10.2:

\(\angle OPQ = \angle OPR\) (OP bisects \(\angle QPR\)).
\(\angle POQ = \angle POR\) (OP bisects \(\angle QOR\)).
Line segment OP, chord QR ka perpendicular bisector hota hai.

💡Tip

Theorem 10.2 ka proof bhi board exams mein frequently pucha jaata hai. Iske saath-saath, iske applications par based numerical problems bhi aate hain.

⭐Remember

Ek external point se circle par sirf do tangents draw ki ja sakti hain. Ye number fixed hai.

Important Theorems aur Unke Applications

Ab tak humne do main theorems padhe hain. Inhe kaise apply karte hain, dekhte hain:

Applications of Theorem 10.1 (Radius \(\perp\) Tangent):

Right-angled triangles: Jab bhi radius aur tangent ka point of contact dikhe, turant \(90^\circ\) ka angle mark kar do. Isse Pythagoras theorem aur trigonometric ratios apply karne mein help milti hai.
Example: Agar ek tangent PQ hai aur radius OQ hai, toh \(\triangle OQP\) ek right-angled triangle hoga at Q.
Finding unknown lengths: Agar radius, tangent ki length, ya center se external point tak ki distance mein se koi do given ho, toh teesri length Pythagoras theorem se nikal sakte hain.

Applications of Theorem 10.2 (Equal Tangent Lengths):

Perimeter of circumscribing figures: Agar ek quadrilateral ya triangle ek circle ko circumscribe karta hai (yaani uski sides circle ko touch karti hain), toh uski perimeter find karne mein ye theorem bahut helpful hota hai.
Example: Agar ABCD ek quadrilateral hai jo circle ko P, Q, R, S par touch karta hai, toh \(AP=AS, BP=BQ, CQ=CR, DR=DS\).
Isse prove kar sakte hain ki \(AB + CD = AD + BC\).
Angles related to tangents: External point se draw kiye gaye tangents ke beech ke angles aur center par banne wale angles ke beech relation establish karne mein helpful.
\(\angle QPR + \angle QOR = 180^\circ\) (Cyclic quadrilateral OQPR).
Finding unknown lengths and angles: Problems mein equal tangent lengths ka use karke equations banate hain aur unknowns solve karte hain.

Special Cases aur Properties:

Parallel Tangents: Ek circle mein maximum do parallel tangents ho sakti hain. Ye hamesha diameter ke end points par hoti hain.
Chord aur Tangent: Agar ek chord ke end points par tangents draw ki jaayen, toh ye tangents equal length ki hoti hain (agar external point same ho) aur center se equal distance par hoti hain.
Concentric Circles: Agar do concentric circles hain aur bade circle ki chord chhote circle ki tangent hai, toh chord ka point of contact chord ko bisect karta hai.

💡Tip

Circumscribing figures (quadrilaterals, triangles) par based questions CBSE ka favourite topic hai. Inmein Theorem 10.2 ka application direct hota hai. Practice these types well!

🧮Formula

Agar \(TP\) aur \(TQ\) external point \(T\) se tangents hain, toh \(TP = TQ\).

Easy (15)

A line that intersects a circle at only one point is called a:
- Secant
- Chord
- Tangent (correct)
- Radius
Why: A tangent is defined as a line that intersects a circle at exactly one point.
The common point of the tangent and the circle is known as the:
- Intersection point
- Point of contact (correct)
- Vertex
- Origin
Why: The point where a tangent touches the circle is called the point of contact.
How many tangents can be drawn to a circle from a point lying inside the circle?
- One
- Two
- Infinitely many
- Zero (correct)
Why: No tangent can be drawn to a circle from a point inside it.
A line that intersects a circle at two distinct points is called a:
- Tangent
- Secant (correct)
- Chord
- Diameter
Why: A secant is a line that cuts a circle at two distinct points.
The maximum number of parallel tangents a circle can have is:
- One
- Two (correct)
- Three
- Infinitely many
Why: A circle can have at most two parallel tangents, at opposite ends of a diameter.
If a line does not intersect a circle at all, it is called a:
- Tangent
- Secant
- Non-intersecting line (correct)
- Chord
Why: A line with no common points with a circle is a non-intersecting line.
The length of the segment of the tangent from an external point P to the point of contact on the circle is called the:
- Radius
- Diameter
- Length of the tangent (correct)
- Chord length
Why: This specific segment is defined as the 'length of the tangent' from that point.
A tangent to a circle is perpendicular to the radius through the point of contact. This statement is:
- Always true (correct)
- Sometimes true
- Never true
- Depends on the circle's size
Why: Theorem 10.1 states that the radius is always perpendicular to the tangent at the point of contact.
A chord of a circle is a segment that connects two points on the circle.
Why: This is the correct definition of a chord.
From an external point, only one tangent can be drawn to a circle.
Why: Two tangents can be drawn to a circle from an external point.
A secant of a circle can be considered a special case of a tangent.
Why: A tangent is a special case of a secant, not the other way around.
A line intersecting a circle in two points is called a ________.
Why: A secant is defined as a line that intersects a circle at two points.
The lengths of tangents drawn from an external point to a circle are ________.
Why: Theorem 10.2 states that tangents from an external point are equal in length.
The tangent at any point of a circle is ________ to the radius through the point of contact.
Why: Theorem 10.1 establishes the perpendicular relationship between a tangent and the radius at the point of contact.
A circle can have ________ point(s) of contact with a tangent line.
Why: By definition, a tangent touches a circle at exactly one point.

Medium (15)

A point P is 13 cm from the center O of a circle. The length of the tangent drawn from P to the circle is 12 cm. Find the radius of the circle.
- 5 cm (correct)
- 7 cm
- 10 cm
- 1 cm
Why: Using Pythagoras theorem, radius = sqrt(13^2 - 12^2) = 5 cm.
If two tangents PA and PB are drawn to a circle with center O from an external point P, and ∠APB = 80°, then what is the measure of ∠POA?
- 50° (correct)
- 40°
- 100°
- 80°
Why: In quadrilateral OAPB, ∠OAP = ∠OBP = 90°. ∠AOB = 180° - 80° = 100°. ΔPOA ≅ ΔPOB, so ∠POA = ∠AOB / 2 = 50°.
A tangent PQ at a point P of a circle of radius 7 cm meets a line through the centre O at a point Q such that OQ = 25 cm. The length of PQ is:
- 18 cm
- 24 cm (correct)
- 32 cm
- 20 cm
Why: Using Pythagoras theorem: PQ = sqrt(OQ^2 - OP^2) = sqrt(25^2 - 7^2) = 24 cm.
In a circle with center O, a chord AB is drawn. If the radius is 10 cm and the perpendicular distance from O to AB is 6 cm, what is the length of the chord AB?
- 8 cm
- 12 cm
- 16 cm (correct)
- 20 cm
Why: Using Pythagoras, half chord length is sqrt(10^2 - 6^2) = 8 cm. Full chord is 16 cm.
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
- 4 cm
- 6 cm
- 8 cm (correct)
- 10 cm
Why: Half chord length = sqrt(5^2 - 3^2) = 4 cm. Full chord length = 8 cm.
Match the following terms with their definitions:
Why: Each term is correctly matched with its definition as given in the chapter.
Match the properties of tangents:
Why: These matches reflect the key theorems and observations about tangents from the chapter.
Match the number of tangents with the position of the point:
Why: The number of tangents depends on whether the point is inside, on, or outside the circle.
What is the term for a line that intersects a circle at exactly one point?
Why: A tangent is defined as a line that intersects a circle at exactly one point.
What is the relationship between the radius and the tangent at the point of contact?
Why: The radius is perpendicular to the tangent at the point of contact (Theorem 10.1).
Arrange the following in increasing order of common points with a circle: (i) Tangent (ii) Non-intersecting line (iii) Secant
Why: Non-intersecting line has 0 points, Tangent has 1 point, Secant has 2 points.
Consider the steps to prove that tangents from an external point are equal. Arrange them in logical order: (i) ΔOAP ≅ ΔOBP (ii) OA ⊥ PA and OB ⊥ PB (iii) PA = PB (iv) Consider triangles OAP and OBP
Why: The proof starts with perpendicularity, then considers triangles, proves congruence, and finally concludes equality of tangents.
Refer to Fig. 10.1 (ii). What is the line PQ called?
- Tangent
- Secant (correct)
- Non-intersecting line
- Chord
Why: Fig. 10.1 (ii) shows a line intersecting the circle at two points, which is a secant.
Observe Fig. 10.1 (iii). What is the line PQ called in this scenario?
- Secant
- Tangent (correct)
- Radius
- Non-intersecting line
Why: Fig. 10.1 (iii) shows a line touching the circle at exactly one point, which is a tangent.
In Fig. 10.7, O is the center of the circle, P is an external point, and PQ and PR are tangents. If ∠QPR = 70°, what is the measure of ∠QOR?
- 70°
- 110° (correct)
- 140°
- 90°
Why: ∠OQP = ∠ORP = 90°. In quadrilateral QORP, ∠QOR + ∠QPR = 180°. So ∠QOR = 180° - 70° = 110°.

Hard (10)

From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is ____ cm.
Why: Using Pythagoras theorem, radius = sqrt(25^2 - 24^2) = 7 cm.
If TP and TQ are two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to ____ degrees.
Why: ∠OPT = ∠OQT = 90°. In quadrilateral OPTQ, ∠PTQ + ∠POQ = 180°. So ∠PTQ = 180° - 110° = 70°.
A quadrilateral ABCD is drawn to circumscribe a circle. If AB = 6 cm, BC = 7 cm, and CD = 4 cm, then the length of AD is ____ cm.
Why: For a circumscribing quadrilateral, AB + CD = BC + AD. So, 6 + 4 = 7 + AD, which gives AD = 3 cm.
In a circle, a chord of length 16 cm is at a distance of 6 cm from the center. The radius of the circle is ____ cm.
Why: Half chord length is 8 cm. Radius = sqrt(8^2 + 6^2) = sqrt(64+36) = sqrt(100) = 10 cm.
Assertion (A): A tangent to a circle intersects the circle at exactly one point. Reason (R): The point where a tangent touches the circle is called the point of contact.
Why: Both statements are individually true, but the reason is a definition, not an explanation for why a tangent intersects at only one point.
Assertion (A): The lengths of tangents drawn from an external point to a circle are equal. Reason (R): The radius drawn to the point of contact is perpendicular to the tangent.
Why: Both statements are true, and the perpendicularity (R) is used in the proof to establish the equality of tangents (A).
A circle is inscribed in a triangle ABC, touching sides BC, CA, and AB at points D, E, and F respectively. If AB = 12 cm, BC = 8 cm, CA = 10 cm, find the length of AF.
- 5 cm
- 6 cm
- 7 cm (correct)
- 4 cm
Why: Let AF=AE=x, BD=BF=y, CD=CE=z. 2(x+y+z) = Perimeter. x+y=12, y+z=8, x+z=10. Solving gives x=7.
Consider a circle with center O and radius 5 cm. A tangent line XY is drawn at point P on the circle. Another point Q is taken on XY such that OQ = 13 cm. What is the length of PQ?
- 8 cm
- 12 cm (correct)
- 10 cm
- 18 cm
Why: OP is radius (5 cm) and OP ⊥ PQ. Using Pythagoras in ΔOPQ: PQ = sqrt(OQ^2 - OP^2) = sqrt(13^2 - 5^2) = 12 cm.
In the given figure, two tangents TP and TQ are drawn to a circle with centre O from an external point T. If ∠PTQ = 60°, then find the measure of ∠POQ.
- 60°
- 90°
- 120° (correct)
- 180°
Why: ∠OPT = ∠OQT = 90°. In quadrilateral OPTQ, ∠POQ + ∠PTQ = 180°. So ∠POQ = 180° - 60° = 120°.
In Fig. 10.13, XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X'Y' at B. What is the measure of ∠AOB?
- 45°
- 60°
- 90° (correct)
- 180°
Why: Join OC. ΔOPA ≅ ΔOCA and ΔOQB ≅ ΔOCB. This leads to ∠POA = ∠COA and ∠QOB = ∠COB. Since POQ is a straight line, 2∠COA + 2∠COB = 180°, so ∠AOB = 90°.