Class 10 / Maths

Class 10 Maths Circles Notes

Notes on tangents to a circle, radius-tangent relation, and lengths of tangents from an external point.

advancedRevision notes

Topic introduction

The Circles chapter in Class 10 focuses mainly on tangents. Most questions use two ideas: the radius is perpendicular to the tangent at the point of contact, and tangents from the same external point are equal.

Key points

  • A tangent touches a circle at exactly one point.
  • The radius drawn to the point of contact is perpendicular to the tangent.
  • Tangents drawn from the same external point to a circle are equal in length.
  • A circle can have two tangents from an external point, one tangent at a point on the circle, and no tangent from a point inside the circle.

Definitions

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

Point of contact: The point where a tangent touches the circle.

External point: A point outside the circle from which two tangents can be drawn.

Formulas

  • Radius at point of contact is perpendicular to tangent.
  • Tangents from an external point are equal.
  • If OP is distance to external point and OT is radius, tangent PT = sqrt(OP^2 - OT^2)

Examples

  1. If PA and PB are tangents from P and PA = 12 cm, then PB = 12 cm.
  2. If OP = 13 cm and radius OT = 5 cm, tangent PT = 12 cm.
  3. No tangent can be drawn from a point inside a circle.

Radius and tangent

If a tangent touches the circle at P and OP is the radius, then OP is perpendicular to the tangent at P.

This creates right triangles, so Pythagoras theorem is often used with tangent questions.

Equal tangents

If PA and PB are tangents from external point P, then PA = PB.

Use this result to form equations when tangent lengths are given as algebraic expressions.

Common mistakes

  • Forgetting that radius and tangent meet at 90 degrees.
  • Using equal tangent property for points that are not the same external point.
  • Drawing a tangent through two points on the circle; that is a secant, not tangent.
  • Forgetting to use Pythagoras theorem in tangent length problems.
  • Writing two tangents from a point on the circle instead of one.

Quick practice

  1. State the tangent-radius theorem.
  2. If PA and PB are tangents from P and PA = 8 cm, find PB.
  3. A radius is 5 cm and distance from center to external point is 13 cm. Find tangent length.
  4. How many tangents can be drawn from a point inside a circle?

Related practice

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