Tangent and Secant Properties

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

Given: A circle with centre O and a tangent XY touching the circle at P.

To Prove: $OP\perp XY$

Proof:

1. Since XY is a tangent, it touches the circle only at P.

2. Point Q lies on the tangent XY and Q ≠ P, so Q lies outside the circle.

3. Therefore, the distance OQ is greater than the radius OP.
   OQ > OP

4. This is true for every point Q on the line XY except P.
   Hence, OP is the shortest distance from O to the line XY.

5. The shortest distance from a point to a line is perpendicular to the line.
   Therefore, OP⊥XY

Statement: The lengths of tangents drawn from an external point to a circle are equal.

Given: A circle with centre O and two tangents PQ and PR drawn from an external point P.

To Prove: PQ = PR

Proof:

1. Join OP, OQ and OR.

2. Radius is perpendicular to the tangent at the point of contact, so

   ∠OQP = ∠ORP = 90^∘

3. OQ = OR (radii of the same circle).

4. OP = OP (common).

5. Therefore, △OQP ≅ △ORP (RHS).

6. Hence, PQ = PR

• A tangent touches a circle at only one point (point of contact).

• The radius through the point of contact is perpendicular to the tangent.

• A line perpendicular to the radius at its endpoint is a tangent to the circle.

• No tangent can be drawn to a circle from a point inside the circle.

• Exactly one tangent can be drawn from a point on the circle.

• Exactly two tangents can be drawn from a point outside the circle.

• From an external point, the two tangents drawn to a circle are equal in length.

• The two tangents from an external point make equal angles at the centre.

• If two circles touch each other, the point of contact lies on the line joining their centres (external and internal touching).