📖 Chapter 10 — Circles (CBSE Class 10 Maths): Detailed notes — tangent to a circle and the two key tangent theorems.
Chapter 10 — Circles
⭕ 1. Tangent & Secant
- A tangent to a circle is a line that touches the circle at exactly one point (the point of contact).
- A secant is a line that intersects a circle at two points.
- A circle can have infinitely many tangents; through a point on the circle there is exactly one tangent.
📐 2. Theorem 1 (Tangent ⊥ Radius)
The tangent at any point of a circle is perpendicular to the radius through the point of contact. (So the radius to the point of contact and the tangent form a right angle → use Pythagoras.)
📏 3. Theorem 2 (Equal Tangents)
The lengths of the two tangents drawn from an external point to a circle are equal. (PA = PB.)
💡 4. Useful Facts
- From an external point, exactly two tangents can be drawn.
- The tangent from an external point, the radius and the line to the centre form a right triangle → tangent = √(d² − r²), where d = distance to centre, r = radius.
- In quadrilateral O-P-T-Q (two tangents TP, TQ): ∠PTQ + ∠POQ = 180°.
🧠 Remember: Tangent touches at 1 point; secant cuts at 2. Tangent ⊥ radius at contact. Tangents from an external point are equal. Tangent length = √(d²−r²). ∠PTQ + ∠POQ = 180°.
⭐ Exam Tips: ① Tangent ⊥ radius → right triangle → Pythagoras. ② Equal tangents from a point. ③ Angle between two tangents (use the quadrilateral). ④ Prove parallel/perpendicular tangent results. ⑤ Tangents at the ends of a diameter are parallel.