Class 10 Maths Notes Chapter 10 — Circles#

Key terms#

  • Secant — a line that meets a circle at two points.
  • Tangent — a line that meets a circle at exactly one point. That point is the point of contact.
  • A tangent is the special case of a secant when its two points of intersection come together.
  • Length of a tangent — the length of the segment from an external point to the point of contact.
  • A line and a circle can have 0, 1 or 2 common points: non-intersecting, tangent, or secant.
  • A circle has infinitely many tangents, one at each point. It can have at most two parallel tangents, at the ends of a diameter.

Tangent and radius#

Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

  • So $\angle OPT = 90°$, where $O$ is the centre, $P$ is the point of contact and $PT$ is the tangent.
  • This gives a right triangle, so use Pythagoras: $OT^2 = OP^2 + PT^2$.
  • The perpendicular to a tangent at its point of contact passes through the centre.
  • Tangents at the two ends of a diameter are parallel.

Number of tangents from a point#

Position of the point Number of tangents
Inside the circle 0
On the circle 1
Outside the circle 2

Tangents from an external point#

Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal. Here, $PA = PB$.

Proof idea: $\triangle OAP \cong \triangle OBP$ by RHS, since $OA = OB$ (radii), $OP$ is common, and $\angle OAP = \angle OBP = 90°$. This also gives:

  • $\angle OPA = \angle OPB$: $OP$ bisects the angle between the tangents.
  • $\angle AOP = \angle BOP$: $OP$ bisects the angle $AOB$.
  • $\triangle PAB$ is isosceles, so $\angle PAB = \angle PBA$.
  • In quadrilateral $OAPB$, the angles at $A$ and $B$ are both 90°, so $$\angle APB + \angle AOB = 180°$$

Results used in exercises#

  • Concentric circles: a chord of the larger circle that touches the smaller one is bisected at the point of contact.
  • Quadrilateral circumscribing a circle: $AB + CD = AD + BC$ (sum of opposite sides are equal). A parallelogram circumscribing a circle is a rhombus.
  • Triangle circumscribing a circle: split each side into the two tangent lengths from its ends, and equate equal tangents.

Common mistakes#

  • Putting the right angle at the external point instead of at the point of contact.
  • Using the length of the tangent as the hypotenuse. The hypotenuse is the line from the centre to the external point.
  • Saying a point inside the circle has one tangent. It has none.

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