Circles

What is a Circle?

A circle is the set of all points in a plane that are at a fixed distance (radius) from a fixed point (centre). In Class 10 CBSE, the focus is on tangents to circles and their properties. This chapter builds on your knowledge of circles from Class 9 (chords, angles, cyclic quadrilaterals) and connects strongly with Pythagoras theorem.

Key Terminology

Tangent: A line that touches a circle at exactly one point. It is the limiting position of a secant when the two intersection points coincide.

Point of Contact: The single point where the tangent touches the circle.

Secant: A line that intersects a circle at two points.

Normal: A line perpendicular to the tangent at the point of contact (it always passes through the centre).

External Point: A point outside the circle from which tangents can be drawn.

Theorem 1: Tangent ⊥ Radius

Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

This is the most important theorem in this chapter. Whenever you see a tangent, immediately draw the radius to the contact point and mark the $90°$ angle. This creates a right triangle, enabling you to use Pythagoras theorem.

Proof idea: If the tangent were not perpendicular to the radius, you could find a point on the tangent closer to the centre than the radius, contradicting the fact that the radius is the shortest distance from the centre to the circle.