Chapter 10 — Circles

NCERT Class 10 Maths Chapter 10 covers Circles, focusing 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 to a circle are equal in length.

Chapter 10 of NCERT Class 10 Mathematics introduces circles and the geometry of tangents. A line can relate to a circle in three ways: no intersection (non-intersecting), two points of intersection (secant), or exactly one point of intersection (tangent). The chapter proves two key theorems: Theorem 10.1 states that the tangent to a circle is perpendicular to the radius at the point of contact, and Theorem 10.2 states that the lengths of tangents drawn from an external point to a circle are equal. The number of tangents possible depends on the point's position — zero from inside, one from on the circle, and exactly two from outside.

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.