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I. Chapter Summary:

This chapter deals with the geometry of circles, focusing on concepts related to tangents. It builds upon the understanding of circles from earlier classes and introduces tangents, number of tangents from a point outside the circle, and properties of tangents drawn from an external point. It strengthens spatial reasoning and geometric proof skills necessary for further studies in geometry and real-world applications like construction, design, and navigation.

II. Key Concepts Covered:

Concept	Description
Circle	A set of all points in a plane that are at a fixed distance from a fixed point (called the centre).
Tangent to a Circle	A line that touches a circle at exactly one point.
Properties of a Tangent	Tangent is perpendicular to the radius at the point of contact.
Number of Tangents from an External Point	Exactly two tangents can be drawn to a circle from a point outside it.
Lengths of Tangents	Tangents drawn from an external point to a circle are equal in length.

III. Important Questions:

(A) Multiple Choice Questions (1 Mark):

1. From a point outside a circle, the number of tangents that can be drawn to the circle is:
   • a) 1
   • b) 2 ✅
   • c) 3
   • d) 0
   • PYQ 2020
2. The angle between a tangent to a circle and the radius drawn to the point of contact is:
   • a) 0°
   • b) 45°
   • c) 90° ✅
   • d) 180°
   • PYQ 2019
3. The length of two tangents drawn from an external point to a circle is always:
   • a) Equal ✅
   • b) Unequal
   • c) One is double the other
   • d) Cannot be determined
4. Which of the following statements is false?
   • a) A tangent to a circle intersects it at two points
   • b) A tangent to a circle is perpendicular to the radius at the point of contact ✅
   • c) Tangents drawn from an external point are equal in length
   • d) Only two tangents can be drawn from a point outside the circle

(B) Short Answer Questions (2/3 Marks):

1. Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°.
2. Prove that the tangents drawn from an external point to a circle are equal in length.
   – PYQ 2022
3. Find the length of each tangent drawn from a point 13 cm away from the centre of a circle of radius 5 cm.
4. Find the number of tangents that can be drawn to a circle from a point lying on it.
   – PYQ 2018

(C) Long Answer Questions (5 Marks):

1. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that $angle PTQ = 2angle OPQ$.
2. Draw a pair of tangents to a circle of radius 4 cm from a point on the exterior which is 6 cm away from the centre.
3. Prove that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.
4. Using the Pythagoras theorem, derive the length of the tangent from a point outside the circle.

(D) HOTS (Higher Order Thinking Skills):

1. A circle is inscribed in a quadrilateral such that it touches all its sides. Prove that the sum of opposite sides of the quadrilateral is equal.
2. In what real-life situations might the property of tangents be used in construction or engineering? Explain with an example.

IV. Key Formulas/Concepts:

Formula/Concept	Explanation
Tangent-Radius Perpendicularity	Tangent to a circle is perpendicular to the radius at the point of contact:
$OT perp PT$
Equal Tangents from External Point	$PA=PB$ where A and B are points of contact
Length of Tangent	$text{Length} = sqrt{d^2 – r^2}$, where $d =$ distance from external point to centre, $r =$ radius of the circle

V. Deleted Portions (CBSE 2025–2026):

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026):

Unit/Chapter	Estimated Marks	Type of Questions Typically Asked
Chapter 10: Circles	3–4 marks	1 Short Answer + 1 MCQ or Diagram-based LAQ

VII. Previous Year Questions (PYQs):

Year	Marks	Question
2018	2	How many tangents can be drawn to a circle from a point lying on the circle?
2019	1	What is the angle between a radius and a tangent at the point of contact?
2020	1	From a point outside a circle, how many tangents can be drawn?
2022	3	Prove that the lengths of tangents drawn from an external point to a circle are equal.

VIII. Real-World Application Examples to Connect with Topics:

Real-World Context	Application of Tangents
Road construction	Designing curves or bends on roads involves tangent lines for smooth turning.
Architecture	Tangent properties help in constructing domes and arches.
Engineering	Tangents are used in designing wheels, pulleys, and gears to ensure perfect fitting.
Navigation	Tangents help determine direction at points where curves meet straight roads.

IX. Student Tips & Strategies for Success:

 Time Management:

• Allocate fixed 30–45 mins daily for Math practice.
• Focus on geometry problems during early morning or high concentration hours.

 Exam Preparation:

• Practice diagrams regularly.
• Memorize key theorems and write proofs clearly.
• Practice PYQs from the last 5 years.

 Stress Management:

• Use visual aids like drawing circles and labeling tangents.
• Practice breathing techniques during revision breaks.

X. Career Guidance & Exploration (Class-Specific):

For Classes 9–10:

• Academic Streams:
  • Science (PCM, PCB), Commerce, Humanities
• Career Paths:
  • Engineering, Architecture, Design, Banking, UPSC, Law
• Foundational Exams:
  • NTSE, Junior Maths Olympiad, NSTSE, SOF IMO

XI. Important Notes:

•  Always refer to the official CBSE website for updated syllabus and sample papers.
•  Focus more on conceptual clarity than memorizing.
•  Consistent revision and diagram-based practice are essential to score well.