Circles — Cheat Sheet

Class X · Mathematics Ch.10 · 8 Panels · Quick Reference

⭕ Line & Circle Basics
A line and a circle can meet in…
LineCommon Points
Non-intersecting0
Tangent1 (point of contact)
Secant2 (a chord)
Key terms: radius = centre to circle; chord = joins two points; diameter = longest chord = 2r.
Tangent is the limiting position of a secant when its two intersection points merge into one.
📍 Tangent to a Circle
A tangent touches the circle at exactly
ONE point → the point of contact.
One per point: through any point ON a circle there is exactly one tangent.
Whole circle lies on one side of the tangent — it touches, never cuts.
Don't confuse: a secant cuts the circle at two points; a tangent only touches at one.
📐 Theorem 1 · Radius ⊥ Tangent
The tangent at any point of a circle is
perpendicular to the radius through
the point of contact.
Why?
Shortest distance from the centre to the tangent line is the perpendicular — and that perpendicular IS the radius.
Converse: a line ⊥ to a radius at its outer endpoint is a tangent. Use it to prove a line is a tangent.
Exam gold: the ∠ between radius and tangent at contact is always 90° — creates a right triangle for calculations.
✌️ Theorem 2 · Equal Tangents
From an external point P:
PA = PB  (the two tangent lengths are equal)
Proof outline
  1. △OAP ≅ △OBP by RHS
  2. OA = OB (radii), OP common, right angles at A, B
  3. ⇒ PA = PB (CPCT)
Also: OP bisects ∠APB and ∠AOB. So ∠OPA = ∠OPB.
🔢 Number of Tangents
Position of PointTangents
Inside the circle0
On the circle1
Outside the circle2 (equal)
Tangent length from external point at distance d:
ℓ = √(d² − r²)  (r = radius)
From: right triangle with hypotenuse d, one leg r → other leg is the tangent.
⭐ Key Results
Circumscribed quadrilateral: AB + CD = BC + DA (sums of opposite sides are equal).
Supplementary angles: ∠ between two tangents + ∠ between the two radii = 180°.
Parallel tangents: the two tangents at the ends of a diameter are parallel.
Perpendicular from centre to a chord bisects the chord.
🌍 Problem Types & Uses
Find Tangent LengthUse ℓ = √(d²−r²) with the right triangle
Prove PA = PBCongruent triangles OAP & OBP
Quadrilateral SidesOpposite side sums equal for circumscribed shapes
Find AnglesUse 90° radius-tangent & angle sum
Real world: pulley belts, bicycle chains and a ball on the ground all form tangents to circles.
🧠 Exam Quick Recall
FactResult
Radius–tangent angle90°
Tangents from ext. pointequal (PA = PB)
Tangent length√(d² − r²)
Point inside / on / out0 / 1 / 2 tangents
Circumscribed quadAB+CD = BC+DA
Common mistakes: forgetting the 90° angle; assuming a secant equals a tangent; mislabelling the point of contact.