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Mathematics Class 10 - Circles Notes

Comprehensive study notes for Class 10 - Circles olympiad preparation

Circles

Welcome to the chapter on Circles for Class 10. In this chapter, you will learn about the properties of circles, different terms related to circles, and important theorems. By the end of this chapter, you will be able to solve problems involving tangents, chords, and angles in circles.

Key Concepts

  • Circle: The set of all points in a plane that are at a fixed distance (radius) from a fixed point (center).
  • Radius: The distance from the center to any point on the circle.
  • Diameter: A chord passing through the center; it is twice the radius.
  • Chord: A line segment joining any two points on the circle.
  • Arc: A part of the circumference of a circle.
  • Sector: The region between two radii and the arc.
  • Segment: The region between a chord and the corresponding arc.
  • Tangent: A line that touches the circle at exactly one point.

Important Properties and Theorems

  • The tangent to a circle is perpendicular to the radius at the point of contact.
  • From a point outside a circle, two tangents can be drawn, and they are equal in length.
  • Equal chords of a circle are equidistant from the center.
  • The angle subtended by an arc at the center is twice the angle subtended at any point on the remaining part of the circle.
  • The sum of the lengths of tangents drawn from an external point to a circle is equal.

Formulas

  • Circumference of a circle: 2πr, where r is the radius.
  • Area of a circle: πr²
  • Length of an arc: (θ/360) × 2πr, where θ is the angle at the center.
  • Area of a sector: (θ/360) × πr²

Practice Questions

  1. Find the circumference of a circle with radius 7 cm.
  2. Calculate the area of a circle with diameter 10 cm.
  3. If the length of a tangent from a point 25 cm from the center of a circle is 24 cm, find the radius of the circle.
  4. Two tangents PA and PB are drawn from an external point P to a circle with center O. If OP = 13 cm and radius = 5 cm, find the length of each tangent.
  5. Prove that the tangents drawn from an external point to a circle are equal in length.

Challenge Yourself

  • Draw a circle and construct two tangents from a point outside the circle. Measure and compare their lengths.
  • If the angle subtended by an arc at the center is 60°, and the radius is 6 cm, find the length of the arc and the area of the sector.

Did You Know?

  • The value of π (pi) is approximately 3.1416, but it is an irrational number and never ends!
  • Circles are used in wheels, clocks, coins, and many real-life objects.

Glossary

  • Radius: Distance from the center to the circle.
  • Diameter: A chord passing through the center (2 × radius).
  • Tangent: A line touching the circle at one point only.
  • Chord: A line segment joining two points on the circle.
  • Arc: A part of the circle's circumference.

Answers to Practice Questions

  1. Circumference = 2π × 7 = 44 cm (approx).
  2. Radius = 5 cm, Area = π × 5² = 78.5 cm² (approx).
  3. Let r be the radius. By the tangent-secant theorem: OP² = r² + length² ⇒ 25² = r² + 24² ⇒ 625 = r² + 576 ⇒ r² = 49 ⇒ r = 7 cm.
  4. Length of tangent = √(OP² - r²) = √(13² - 5²) = √(169 - 25) = √144 = 12 cm.
  5. Proof: Tangents from an external point to a circle are equal in length (use the tangent-secant theorem or congruent triangles).

Circles are everywhere! Practice problems to master their properties and applications.

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