🎯 Class 10 Maths Chapter 10: Circles – Notes

 

🌀 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.

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🔵 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.

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✏️ 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.

   $$OP \perp AB$$

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

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

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🧮 2. Number of Tangents from a Point

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🔺 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:

$$PA = PB$$

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

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🧠 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.

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📝 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:

$$OA^2 + AP^2 = OP^2 \Rightarrow r^2 + 6^2 = 10^2 \Rightarrow r^2 = 100 - 36 = 64 \Rightarrow r = 8 \text{ cm}$$

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📌 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.

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📥 Download Chapter 10 PDF Notes: Coming Soon

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📚 Next Chapter: Chapter 11 – Constructions »