What this chapter is about
This chapter explores the relationship between circles and lines, particularly tangents. You already know basic circle concepts from earlier classes — radius, diameter, chord, arc, sector. Now you study how a line can touch a circle at exactly one point and what special properties this creates.
The main focus is on tangents to circles. A tangent is a line that meets a circle at precisely one point, called the point of contact. You will learn why a tangent is always perpendicular to the radius at that point, and how to work with tangents drawn from an external point to a circle.
After this chapter, you should be able to prove results about tangents, calculate lengths of tangent segments, and solve problems involving circles with tangents drawn from external points. These ideas connect geometry with coordinate geometry and trigonometry in later work.
Key ideas
- A tangent to a circle is a line that touches the circle at exactly one point; a secant is a line that cuts the circle at two points.
- The tangent at any point on a circle is perpendicular to the radius drawn to that point of contact.
- From a point outside a circle, exactly two tangents can be drawn to the circle.
- The lengths of the two tangent segments drawn from an external point to a circle are equal.
- The line joining an external point to the centre of the circle bisects the angle between the two tangents from that point.
- If two tangents are drawn from an external point, the line joining the external point to the centre passes through the midpoint of the chord joining the two points of contact.
- The perpendicular from the centre of a circle to a chord bisects the chord (useful when tangents and chords appear together in problems).
Formulas and facts to remember
Tangent perpendicular to radius: If PT is a tangent to a circle with centre O at point T, then OT ⊥ PT. Meaning: The radius and tangent meet at a right angle.
Equal tangent lengths: If PA and PB are tangents from external point P to a circle with centre O, touching at A and B, then PA = PB. Meaning: Both tangent segments from the same outside point have the same length.
Angle between tangents: If PA and PB are tangents from P, then ∠APB and ∠AOB are supplementary, meaning ∠APB + ∠AOB = 180°. Meaning: The angle at the external point and the angle at the centre (formed by the two radii to the contact points) add up to 180°.
Right triangle formed: Triangle OAP (where O is centre, A is contact point, P is external point) is right-angled at A. Meaning: You can apply Pythagoras theorem: OP² = OA² + PA².
Worked examples
Example 1
A circle has centre O and radius 5 cm. A point P is 13 cm from O. Find the length of the tangent from P to the circle.
Solution: Let the tangent from P touch the circle at point T. Since tangent is perpendicular to radius at contact point, angle OTP = 90°. In right triangle OTP: OP² = OT² + PT² 13² = 5² + PT² 169 = 25 + PT² PT² = 144 PT = 12 cm
The tangent length is 12 cm.
Example 2
Two tangents PA and PB are drawn from an external point P to a circle with centre O. If ∠APB = 60°, find ∠AOB.
Solution: We know that ∠APB + ∠AOB = 180° (tangents from external point property). 60° + ∠AOB = 180° ∠AOB = 120°
The angle at the centre is 120°.
Example 3
From a point P outside a circle with centre O and radius 4 cm, two tangents PA and PB are drawn. If PA = 3 cm, find the distance OP.
Solution: Since PA is a tangent and OA is a radius, angle OAP = 90°. In right triangle OAP: OP² = OA² + PA² OP² = 4² + 3² OP² = 16 + 9 = 25 OP = 5 cm
The distance from P to the centre is 5 cm.
Common mistakes
Forgetting that tangent meets radius at 90° → Always draw the radius to the contact point and mark the right angle before solving.
Assuming tangent passes through the centre → The tangent never passes through the centre; it only touches the circle at one point on the circumference.
Using the wrong triangle for Pythagoras → The right angle is at the point of contact, not at the external point or the centre.
Thinking unequal tangent lengths from an external point → Both tangents from the same external point are always equal in length.
Confusing tangent with chord or secant → A tangent touches at one point only; a chord joins two points on the circle; a secant cuts through two points.
Quick revision
- Tangent touches circle at exactly one point; perpendicular to radius there.
- From an external point, two tangents can be drawn, both of equal length.
- Use Pythagoras in the right triangle formed by radius, tangent and line to centre.
- ∠APB + ∠AOB = 180° when PA and PB are tangents from external point P.
- The radius to the contact point is your key line — draw it first in every problem.