Login / Register
Follow Us
0
0
Register Free!
Register Free!
Online Mock Tests & Exam Preparation | Target Exams
Select Medium / माध्यम चुनें:
Eng (English) Hindi (हिन्दी)
CBSE • Class X • Mathematics • Ch 10
Estimated Time: 45 Mins
Study Progress: In Progress

Circles

In Class 10 Mathematics, "Circles" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

Take Chapter CBT Test View All Chapters
Overview Core Concepts Summary Self Check Take MCQ Test

⚪ Have You Ever Wondered?

Why can you draw infinitely many secant lines across a circle, but exactly two tangents from any external point? Circle geometry underpins planetary o...

Why can you draw infinitely many secant lines across a circle, but exactly two tangents from any external point? Circle geometry underpins planetary orbits and mechanical gear tooth design.

Why This Chapter Matters

In Class 10 Mathematics, "Circles" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

Before You Begin (Prerequisites)

  • Circles, chords, radii from Class 9.
  • Right triangles and Pythagoras theorem.
  • Congruence of triangles.

What You Will Learn (Core Objectives)

  • Define a Tangent to a circle as a line intersecting the circle at exactly one point.
  • Prove Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact ($OP \perp AB$).
  • Prove Theorem 2: The lengths of tangents drawn from an external point to a circle are equal ($AP = BP$).
  • Apply the tangent theorems to solve angle and length calculations.
  • Prove geometric circle theorems involving inscribed and circumscribed quadrilaterals.

Chapter Roadmap & Progression

1 1. Tangents and The Perpendicular R...
2 2. Tangents from an External Point
3 3. Tangents Subtend Equal Angles

Complete Concept Guide (100% Curriculum Coverage)

1. Tangents and The Perpendicular Radius

A Tangent touches a circle at exactly one point: the Point of Contact.
Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact: $$\mathbf{OP \perp AB}$$

2. Tangents from an External Point

From an external point $P$, exactly two tangents $PA$ and $PB$ can be drawn to a circle.
Theorem 2: The lengths of tangents drawn from an external point to a circle are equal: $$\mathbf{PA = PB}$$
Proved via RHS congruence on $\triangle OPA$ and $\triangle OPB$!

3. Tangents Subtend Equal Angles

The two tangents subtend equal angles at the center ($\angle AOP = \angle BOP$) and are equally inclined to the line segment joining the center to the point ($\angle APO = \angle BPO$).

Visual Learning & Conceptual Map

Circles Master Matrix

Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture

1. Tangents and The Perpendicular Radius • 2. Tangents from an External Point

Chapter Summary & 10 Key Takeaways

Takeaway 1
Tangent Definition: Line touching circle at strictly one point of contact.
Takeaway 2
Perpendicular Radius: Radius to point of contact is perpendicular to tangent ($90^\circ$).
Takeaway 3
Equal Tangents: $PA = PB$ from any external point $P$.
Takeaway 4
Supplementary Center: Angle between two tangents and angle subtended at center sum to $180^\circ$.
Takeaway 5
Circumscribed Quadrilateral: Opposite sides sum equally: $AB + CD = AD + BC$.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
A tangent $PQ$ at a point $P$ of a circle of radius 5 cm meets a line through the center $O$ at a point $Q$ so that $OQ = 12\text{ cm}$. Find length $PQ$.
Reveal Answer & Explanation
Answer: Since $OP \perp PQ$, in right $\triangle OPQ$: $PQ^2 = OQ^2 - OP^2 = 12^2 - 5^2 = 144 - 25 = 119 \implies PQ = \sqrt{119}\text{ cm}$.
PQ = sqrt(119) cm.
2
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Reveal Answer & Explanation
Answer: The radius at each end of the diameter is perpendicular to the tangents ($90^\circ$ each). These form equal alternate interior angles ($90^\circ = 90^\circ$), which guarantees that the two tangents are parallel.
Alternate interior angles equal 90 degrees.
3
If tangents $PA$ and $PB$ from point $P$ to circle with center $O$ are inclined at $80^\circ$, find $\angle POA$.
Reveal Answer & Explanation
Answer: Angle between tangents ($80^\circ$) and central angle $\angle AOB$ are supplementary $\implies \angle AOB = 180 - 80 = 100^\circ$. Line $OP$ bisects $\angle AOB$, so $\angle POA = \frac{100^\circ}{2} = 50^\circ$.
Angle POA = 50°.
4
Prove that in two concentric circles, the chord of the larger circle which touches the smaller circle is bisected at the point of contact.
Reveal Answer & Explanation
Answer: The radius of the smaller circle to the point of contact is perpendicular to the tangent chord. By Class 9 theorem, the perpendicular from the center of a circle to a chord bisects the chord. Hence the chord is bisected.
Perpendicular from center bisects chord.
5
How many tangents can a circle have from a point lying inside the circle?
Reveal Answer & Explanation
Answer: Zero tangents (any line through an interior point is a secant intersecting in two points).
Zero tangents from inside.
Finished Studying This Chapter?
READY TO PRACTICE?

Timed CBT Practice Tests (Exam Simulator)

Put your concepts to the test with official curriculum-aligned Foundation and Advanced practice tests. Get instant accuracy scores, time metrics, and step-by-step verified explanations.

Foundation Test
15 MCQs • Core Concepts
Advanced Test
15 MCQs • HOTS & Traps

All Class 10 Mathematics Chapters

AI Study Friend

Instant Concept Tutor

Have a doubt in Circles? Ask our AI study tutor for rapid explanations or customized quizzes.