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Class 10 Mathematics Exercise 10.1 NCERT Solutions Olympiad Board Exam

Chapter 10 — Circles

Step-by-step NCERT solutions with stress–strain analysis and exam-oriented hints for Boards, JEE & NEET.

📋4 questions
⏱Ideal time: 30-40 min
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Q1
NUMERIC3 marks

How many tangents can a circle have?

Conceptual Theory

A tangent to a circle is a straight line that touches the circle at exactly one point.

The point where the tangent touches the circle is called the point of contact.

A fundamental geometric property states:

The tangent at any point of a circle is perpendicular to the radius drawn to the point of contact.

Since a circle is made up of infinitely many points along its circumference, each point can have exactly one tangent.

Solution Roadmap
  • Understand definition of tangent
  • Recognize number of points on a circle
  • Relate each point to a unique tangent
  • Conclude total number of tangents
O P Tangent
Step-by-Step Solution

Step 1: A tangent to a circle touches it at exactly one point.

Step 2: A circle is a continuous curve consisting of infinitely many points.

Step 3: At each point on the circle, exactly one tangent can be drawn.

Step 4: Therefore, corresponding to infinitely many points, there are infinitely many tangents.

Final Answer: A circle can have infinitely many tangents.

Exam Significance
  • This is a fundamental conceptual question frequently asked in CBSE board exams (1-mark direct question).
  • Forms the base for proving tangent-related theorems in later questions.
  • Important for competitive exams like NTSE, Olympiads, and basic geometry sections of JEE foundation.
  • Helps in understanding advanced concepts like lengths of tangents and tangent properties.
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1 / 4  ·  25%
Q2 →
Q2
NUMERIC3 marks

Fill in the blanks :
(i) A tangent to a circle intersects it in ___________ point (s).
(ii) A line intersecting a circle in two points is called a _____________.
(iii) A circle can have __________ parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called ___________.

Conceptual Theory

Tangent: A line that touches a circle at exactly one point.

Secant: A line that cuts a circle at two distinct points.

Parallel Tangents: A circle can have at most two tangents that are parallel to each other.

Point of Contact: The exact point where a tangent touches the circle.

Solution Roadmap
  • Recall definitions of tangent and secant
  • Understand intersection behaviour of lines with circle
  • Visualize parallel tangents geometrically
  • Identify correct terminology
P Tangent
A B Secant
T₁ T₂
Step-by-Step Justification

(i) A tangent touches the circle at exactly one point.
Hence, the number of intersection points is: \[ 1 \]

(ii) A line that cuts a circle at two distinct points is called a secant.
Therefore, the correct term is: \[ \text{Secant} \]

(iii) For a given circle, only two parallel tangents can exist—one on each opposite side.
Hence: \[ 2 \]

(iv) The point where the tangent touches the circle is called: \[ \text{Point of Contact} \]

Final Answers

(i) One
(ii) Secant
(iii) Two
(iv) Point of Contact

Exam Significance
  • Direct 1-mark conceptual question in CBSE board exams
  • Frequently used to test basic definitions in geometry
  • Forms foundation for tangent theorems and constructions
  • Highly relevant for NTSE, Olympiad, and basic aptitude geometry
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2 / 4  ·  50%
Q3 →
Q3
NUMERIC3 marks

A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :
(A) 12 cm (B) 13 cm (C) 8.5 cm (D) 119 cm.

O P Q

Q3. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q such that OQ = 12 cm. Find PQ.

Conceptual Theory

The tangent at any point of a circle is perpendicular to the radius at the point of contact.

Therefore, \(\angle OPQ = 90^\circ\), and triangle OPQ is a right-angled triangle.

Solution Roadmap
  • Identify right triangle OPQ
  • Apply Pythagoras theorem
  • Substitute given values
  • Solve step-by-step
Step-by-Step Solution

Step 1: Given radius: \[ OP = 5 \text{ cm}, \quad OQ = 12 \text{ cm} \]

Step 2: Since radius is perpendicular to tangent: \[ \angle OPQ = 90^\circ \]

Step 3: Apply Pythagoras theorem in right triangle OPQ: \[ OQ^2 = OP^2 + PQ^2 \]

Step 4: Substitute values: \[ 12^2 = 5^2 + PQ^2 \]

Step 5: Simplify: \[ 144 = 25 + PQ^2 \]

Step 6: Solve for \(PQ^2\): \[ PQ^2 = 144 - 25 = 119 \]

Step 7: Take square root: \[ PQ = \sqrt{119} \]

Final Answer: \(\sqrt{119}\) cm

Correct Option: (D) \(\sqrt{119}\) cm

Exam Significance
  • Very common 2-mark question in CBSE board exams
  • Tests application of tangent-radius perpendicular property
  • Direct use of Pythagoras theorem in geometry
  • Important for NTSE, Olympiad, and basic coordinate geometry concepts
  • Frequently used as a base for higher-level tangent length problems
Common Mistake Alert

Students often incorrectly write: \[ OP^2 = OP^2 + PQ^2 \] which is wrong.

Correct relation is: \[ OQ^2 = OP^2 + PQ^2 \]

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3 / 4  ·  75%
Q4 →
Q4
NUMERIC3 marks

Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.

Q4. Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.

Conceptual Theory

A tangent touches the circle at exactly one point, whereas a secant intersects the circle at two distinct points.

For a fixed direction (parallel lines), shifting the line relative to the circle changes its nature:

  • If the distance from the centre equals radius → Tangent
  • If the distance is less than radius → Secant
Construction Roadmap
  • Draw a circle with centre O
  • Draw a reference line (given direction)
  • Draw a parallel line touching the circle → Tangent
  • Draw another parallel line cutting the circle → Secant
O PQ (Tangent) P RS (Secant) R S
Step-by-Step Construction

Step 1: Draw a circle with centre \(O\) and any convenient radius.

Step 2: Draw a straight line \(l\) (this will define the direction for parallel lines).

Step 3: Draw a line \(PQ\) parallel to \(l\) such that it touches the circle at exactly one point.

Step 4: This line \(PQ\) is the tangent.

Step 5: Draw another line \(RS\), also parallel to \(l\), but passing through the circle.

Step 6: This line intersects the circle at two points \(R\) and \(S\), hence it is a secant.

Final Result

Line \(PQ\) is a tangent and line \(RS\) is a secant to the circle, and both are parallel to the given line.

Exam Significance
  • Important construction-based question for CBSE board practical geometry
  • Tests conceptual clarity between tangent and secant
  • Frequently asked in viva and internal assessments
  • Builds foundation for locus and distance-based reasoning
  • Useful in competitive exams for visual reasoning problems
Common Error Alert

Students often draw both lines either fully outside or fully intersecting the circle.

Remember:
Tangent → exactly one point
Secant → exactly two points

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4 / 4  ·  100%
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Chapter Complete!

All 4 solutions for Circles covered.

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NCERT Class X · Mathematics

Chapter 10 · Circles

Exercise 10.1 — A complete AI-powered learning engine with proofs, solver, interactive modules, and concept-building questions

📐 2 Core Theorems
🧠 AI Step Solver
⚡ 4 Interactive Modules
📚 18 Practice Questions
📖

Core Concepts at a Glance

Click any card to expand detailed properties

CONCEPT 01

Circle & Its Parts

A circle is the locus of all points equidistant from a fixed point called the centre. The fixed distance is the radius.

  • Chord: Line segment joining two points on the circle
  • Diameter: Longest chord passing through the centre
  • Arc: Part of the circle between two points
  • Sector: Region between two radii and an arc
  • Segment: Region between a chord and its arc
CONCEPT 02

Tangent to a Circle

A tangent to a circle is a line that touches the circle at exactly one point. That unique point is called the point of tangency.

  • A tangent never intersects the circle — it merely "grazes" it
  • The tangent is perpendicular to the radius at the point of tangency
  • Only one tangent can be drawn at any point on a circle
  • Tangents from an external point are equal in length
CONCEPT 03

Secant vs. Tangent

A secant is a line that intersects a circle at two distinct points, while a tangent touches at exactly one point.

  • A tangent can be considered a limiting case of a secant
  • As the two intersection points of a secant coincide, it becomes a tangent
  • Secants produce two chords (one inside, one outside — useful in power-of-a-point theorem)
CONCEPT 04

Number of Tangents from a Point

The number of tangents from a point depends on the point's position relative to the circle.

  • Point inside the circle → 0 tangents possible
  • Point on the circle → exactly 1 tangent
  • Point outside the circle → exactly 2 tangents
  • Both tangents from an external point are always equal in length
CONCEPT 05

Theorem 10.1 (Tangent ⊥ Radius)

The tangent at any point of a circle is perpendicular to the radius through the point of tangency.

  • ∠OPT = 90° where O = centre, P = point of tangency, T = any point on tangent
  • Proof uses: OQ > OP for all other points Q on the tangent line
  • OP is the shortest distance from centre to the tangent line
  • Converse also holds: A line perpendicular to a radius at its outer end is a tangent
CONCEPT 06

Theorem 10.2 (Equal Tangents)

The lengths of two tangents drawn from an external point to a circle are always equal.

  • If PA and PB are tangents from P, then PA = PB
  • Proof: △OAP ≅ △OBP by RHS congruence (OA=OB=r, OP=OP, ∠OAP=∠OBP=90°)
  • ∠OPA = ∠OPB — the line OP bisects the angle between the two tangents
  • OP also bisects the chord AB (the chord of contact)
⚖️

Theorems in Full

The two key theorems of Exercise 10.1

Theorem 10.1

Tangent is Perpendicular to Radius

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Proof Sketch: Let O be the centre, P the point of tangency, and XY the tangent. Suppose the radius OP is not perpendicular to XY. Then there exists a point Q on XY such that OQ ⊥ XY, making OQ < OP. But Q lies outside the circle (since it's on the tangent), so OQ > radius = OP — a contradiction. Hence OP ⊥ XY, i.e., ∠OPY = 90°.

★ Contradiction method used in proof

Theorem 10.2

Tangents from External Point are Equal

The lengths of the two tangent segments from an external point to a circle are equal.

Proof: Let PA and PB be tangents from external point P to circle with centre O. In △OAP and △OBP:
• OA = OB (radii)
• OP = OP (common side)
• ∠OAP = ∠OBP = 90° (tangent ⊥ radius)
By RHS congruence: △OAP ≅ △OBP → PA = PB (CPCT).

★ RHS Congruence rule applied

📐

Formula Reference Sheet

All key relationships for Circle Chapter 10

Tangent Length from External Point

If P is an external point at distance d from centre O of a circle with radius r, the length of the tangent PT is given by Pythagoras theorem applied to right △OTP.

PT = √(d² − r²)

Pythagoras in △OTP

Since ∠OTP = 90° (tangent ⊥ radius), the triangle OTP is right-angled at T. The three sides satisfy the Pythagorean relation.

OP² = OT² + PT²

Angle at Point of Tangency

The radius drawn to the point of tangency is always perpendicular to the tangent line at that point. This is Theorem 10.1.

∠OPT = 90°

Equal Tangents (Theorem 10.2)

If PA and PB are two tangents from external point P to a circle, both tangents are equal in length regardless of where P is placed outside.

PA = PB

Angle Bisection Property

The line joining the external point P to the centre O bisects the angle between the two tangents (∠APO = ∠BPO) and also bisects the chord of contact AB.

∠APO = ∠BPO

Angle Between Tangent and Chord

Tangent-chord angle equals the inscribed angle in the alternate segment. This is the Alternate Segment Theorem (also called Tangent-Chord angle).

∠TAB = ∠ACB

Perimeter of Triangle with Incircle

If a circle of radius r is inscribed in a triangle with sides a, b, c and semi-perimeter s, then the area of the triangle relates to r and s.

Area = r × s

Tangent & Secant Power

If from external point P, a tangent of length t and a secant with external segment a and full chord length a+b are drawn:

t² = a(a+b)
💡

Tips, Tricks & Shortcuts

Smart strategies for solving circle problems fast

🎯

Always Mark the 90°

The moment you spot a tangent touching a circle, immediately mark a 90° angle between the tangent and radius at the point of contact. This unlocks Pythagoras and many angle-related properties.

⚖️

Equal Tangents = Isoceles Triangle

Two tangents from an external point to a circle are equal, so △OAP ≅ △OBP. The triangle formed by the two tangents and the chord of contact is always isoceles.

🔺

Use RHS Congruence

For problems involving tangents from an external point, use RHS (Right angle – Hypotenuse – Side) congruence. OA=OB (radii) + right angles + common hypotenuse OP almost always applies.

📏

Pythagoras Shortcut

To find tangent length from external point P: measure the distance from P to the centre (OP), and subtract the radius squared: PT = √(OP² − r²). This is almost always tested numerically.

🔄

Number of Tangents Trick

Memorise: 0 (inside), 1 (on), 2 (outside). For any question asking "how many tangents can be drawn?", first determine the point's position using its distance from the centre vs. the radius.

🧩

Incircle Perimeter Trick

For a quadrilateral circumscribed about a circle (circle touches all 4 sides), AB + CD = BC + DA. This elegant property follows directly from equal tangents — a favourite exam question!

🔍

Converse Theorem

If a line is perpendicular to a radius at its endpoint on the circle, it is a tangent. Use this converse to prove lines are tangents — write "by converse of Theorem 10.1" in solutions.

✍️

Angle Chase Strategy

In problems with tangents and triangles, start by listing all known angles (including the 90° at tangency), then use angle-sum in triangles to find unknowns systematically.

⚠️

Common Mistakes to Avoid

Errors students frequently make — and how to fix them

✗

Confusing Tangent with Chord

Students often incorrectly treat a tangent as a chord. Remember: a chord connects two points on the circle (inside), while a tangent only touches at one point and lies entirely outside (except at that point).

A tangent has exactly one point common with the circle. If two points are on the circle, it's a chord/secant, not a tangent.
✗

Forgetting the 90° Angle

Not marking ∠OPT = 90° at the point of tangency is the most common error. Without this, Pythagoras and most proofs cannot proceed. Always state "radius ⊥ tangent at point of tangency" as a reason.

In every tangent problem, the very first step must be: mark ∠OPT = 90° and state Theorem 10.1 as the reason.
✗

Incorrect Number of Tangents from a Point Inside

Students sometimes write "1 tangent" for a point inside the circle. No tangent can be drawn from a point inside a circle — any line through it will always intersect the circle at two points.

0 tangents from an interior point. This is because OQ < r for any point Q inside, so the line will always cross the circle.
✗

Wrong Congruence Rule Cited

Using SSA (which is not a valid congruence rule) instead of RHS when proving △OAP ≅ △OBP. Since ∠OAP = ∠OBP = 90°, OP = OP (hypotenuse), OA = OB (radii) — this is RHS, not SSS or SAS.

When both triangles have a right angle, equal hypotenuse, and one equal side, always cite RHS (Right-Hypotenuse-Side) congruence.
✗

Applying PT = √(d² + r²) Instead of √(d² − r²)

A common arithmetic error — adding r² instead of subtracting it. Since OP is the hypotenuse, OP² = OT² + PT², so PT² = OP² − OT² = d² − r². The sign must be subtraction.

Draw the right triangle first. The centre-to-external-point distance is always the hypotenuse. Tangent and radius are the two legs.
🤖

AI Step-by-Step Solver

Type any circle problem or select a sample question

Your Question
Tangent Length
Prove Equal Tangents
Angle Proof
Circumscribed Quad
Find ∠APB
Solution

Step-by-Step Solution

    📝

    Concept-Building Questions

    18 original questions with full step-by-step solutions, organised by concept

    🎮

    Interactive Learning Modules

    Choose a module to start

    🎯

    Quick-Fire Quiz

    10 MCQ questions covering all concepts of Exercise 10.1. Instant feedback with explanation.

    ▶ Start Quiz →
    ✏️

    Fill in the Blanks

    Complete the statements, theorems, and proofs. Tests recall and precise language.

    ▶ Start →
    🔗

    Concept Matching

    Match theorems, properties, and diagrams to their correct statements. Visual memory builder.

    ▶ Start →
    ⚖️

    True or False

    Rapidly test misconceptions and edge cases. 8 tricky statements about tangents and circles.

    ▶ Start →
    🎨

    Visual Explorer

    Interactive diagram — drag & explore circle properties

    Adjust the sliders to explore how tangent length changes with distance from the centre.
    📚
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    NCERT Class 10 Maths Ch 10 Ex 10.1 Circles Solutions (Step-by-Step)
    NCERT Class 10 Maths Ch 10 Ex 10.1 Circles Solutions (Step-by-Step) — Complete Notes & Solutions · academia-aeternum.com
    The Solutions to the Textbook Exercises of NCERT Mathematics Class X Chapter 10 – “Circles” are designed to provide students with clear, logically structured, and exam-ready explanations for every problem prescribed in the syllabus. This chapter plays a crucial role in strengthening deductive reasoning, as most questions require a sound understanding of definitions, theorems, and their correct application through well-drawn diagrams and step-wise proofs. These solutions focus on developing…
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