Circles

This chapter studies the different ways a line can meet a circle and, in particular, the tangent to a circle — a line that touches the circle at exactly one point. It proves the two fundamental tangent theorems: the tangent at any point of a circle is perpendicular to the radius through the point of contact (Theorem 10.1), and the lengths of the two tangents drawn from an external point are equal (Theorem 10.2). It also develops the length-of-tangent formula $\sqrt{d^2-r^2}$, the number of tangents from a point, and a rich set of board applications — figures circumscribing a circle (quadrilaterals, triangles, parallelograms) and the angle relations between tangents and the centre. It is a short but very high-scoring chapter of the Class 10 Board syllabus.