Chapter 10 — Circles

Chapter 10 of the Class 10 Maths NCERT textbook, "Circles", focuses on tangents to a circle — proving that the tangent at any point is perpendicular to the radius at the point of contact, and that the two tangents drawn from an external point are equal in length.

• How a line meets a circle — A line can miss a circle entirely, cut it at two points as a secant, or touch it at exactly one point as a tangent. A tangent is the limiting case where a secant's two crossings merge into one.
• Tangent and radius — At the point where a tangent touches, it is always perpendicular to the radius drawn to that point. This right-angle relationship is the key property behind most tangent problems.
• Tangents from an outside point — From a point outside a circle exactly two tangents can be drawn, and they have equal length. From a point on the circle there is one tangent, and from inside, none.

1. 01A tangent to a circle intersects it at exactly one point, called the point of contact; a secant intersects at two points.
2. 02Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact (OP ⊥ XY).
3. 03Theorem 10.2: The lengths of the two tangents drawn from an external point to a circle are equal (PQ = PR).
4. 04From a point inside a circle, no tangent can be drawn; from a point on the circle, exactly one tangent exists; from a point outside, exactly two tangents can be drawn.
5. 05The centre of a circle lies on the angle bisector of the angle formed between the two tangents drawn from an external point.
6. 06A tangent is a limiting case of a secant — when the two intersection points of the secant coincide at one point.