Circles

Chapter at a Glance

• A tangent to a circle intersects the circle at exactly one point (the point of contact).
• A secant intersects the circle at two points; a non-intersecting line has none.
• A tangent is the limiting case of a secant when the endpoints of the chord coincide.
• The tangent at any point is perpendicular to the radius through the point of contact.
• From a point inside the circle: no tangent; on the circle: exactly one tangent; outside the circle: exactly two tangents.
• Lengths of the two tangents from an external point to a circle are equal.
• The two tangents from an external point and the line joining the centre to the external point satisfy specific angle and bisector relations.

Definitions, Theorems and Results

Definitions (as framed in NCERT): - Tangent: a line that intersects the circle at only one point. - Point of contact: the common point of the tangent and the circle. - Length of the tangent from an external point P: the length of the segment from P to the point of contact.

Theorems (both require proof in the exam): - Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact. - Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.

Other standard results stated in the chapter: - There is one and only one tangent at any point on a circle. - There are no tangents from an interior point, exactly one from a point on the circle, and exactly two from an exterior point. - The line from centre to external point bisects the angle between the two tangents. - Perpendicular from the centre to a chord bisects the chord (used when tangent is perpendicular to radius).

Formula Sheet

No algebraic formulas are given in the chapter; relations are geometric. The only quantitative relations derived are:

Solved-Example Patterns

Type 1: Prove tangent ⊥ radius or use it to show bisection
Join centre to point of contact; apply Theorem 10.1 to obtain right angle; then use “perpendicular from centre bisects chord”.

Type 2: Prove two tangents from external point are equal
Join centre to external point and points of contact; show two right triangles congruent by RHS (or apply Pythagoras directly).

Type 3: Find length of tangent or chord

Form right triangle with radius and tangent; apply Pythagoras or similarity after locating intersection of line of centres with chord.

Type 4: Angle chasing with two tangents

Use isosceles triangle formed by equal tangents + right angles at points of contact to relate angles at external point and centre.

Type 5: Prove parallel tangents or supplementary angles
Apply Theorem 10.1 at both ends of diameter or join centre and use vertically opposite/linear pair angles.

Common Mistakes and Exam Pitfalls

• Forgetting to state that the point Q lies outside the circle when proving Theorem 10.1.
• Missing the RHS congruence step or the “common OP” in Theorem 10.2 proof.
• Assuming more than two tangents from an external point or drawing a tangent from an interior point.
• Omitting the right-angle mark at the point of contact when using Theorem 10.1.
• Not marking equal tangent lengths or the isosceles triangle when angles are required.
• Using numerical values without units or forgetting to verify that the calculated length satisfies the triangle inequality in the figure.