🌀 Introduction

A circle is one of the most fundamental shapes in geometry. In earlier classes, we learned about its basic properties—radius, diameter, chord, and arc. In this chapter, we explore a key concept: tangents to a circle, and learn how to solve problems involving tangents from a point outside a circle.


🔵 What is a Circle?

A circle is the set of all points in a plane that are equidistant from a fixed point, called the centre.

  • Radius (r): Distance from the centre to any point on the circle.

  • Diameter (d): Twice the radius.

  • Chord: A line segment joining any two points on the circle.

  • Arc: A part of the circle's circumference.

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


✏️ 1. Tangents to a Circle

A tangent is a line that touches the circle at only one point and does not cut through it.

🔹 Properties of a Tangent:

  1. A tangent is perpendicular to the radius at the point of contact.

    OPABOP \perp AB

    (Where O = center, P = point of contact, AB = tangent)

  2. From an external point, exactly two tangents can be drawn to a circle.


🧮 2. Number of Tangents from a Point

Position of the PointNumber of Tangents
Point on the circle1 tangent
Point outside the circle2 tangents
Point inside the circle0 tangents

🔺 3. Length of Tangents from an External Point

If two tangents are drawn from an external point to a circle:

  • The lengths of the tangents are equal.

  • Triangles formed are congruent.

Let:

  • O be the center of the circle.

  • P be a point outside the circle.

  • PA and PB be tangents to the circle.

Then:

PA=PBPA = PB

And triangles △OPA and △OPB are congruent by RHS congruency.


🧠 4. Common Theorems (Class 10 Level)

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

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

These theorems are used frequently to prove statements and solve construction-based problems.


📝 Sample Questions

Q1. Two tangents TP and TQ are drawn to a circle with center O from an external point T. Prove that ∠PTQ = 2∠OPQ.

✍️ Hint: Use congruent triangles and properties of circle geometry.

Q2. From a point 10 cm away from the center of a circle, a tangent of 6 cm is drawn. Find the radius.

Solution:
Use Pythagoras theorem in triangle OAP:

OA2+AP2=OP2r2+62=102r2=10036=64r=8 cmOA^2 + AP^2 = OP^2 \Rightarrow r^2 + 6^2 = 10^2 \Rightarrow r^2 = 100 - 36 = 64 \Rightarrow r = 8 \text{ cm}

📌 Key Points to Remember

  • Tangents are always perpendicular to the radius.

  • A point outside a circle has exactly two tangents.

  • Tangent segments from a point outside a circle are equal in length.

  • Use congruence and Pythagoras theorem in problems.


📥 Download Chapter 10 PDF Notes: Coming Soon

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