Ever wondered why a line just kisses a circle at one point?
💡 In Simple Words: A tangent is a straight line that touches a circle at exactly one point. A chord is any line segment whose ends both lie on the circle. Both have neat tricks that help you solve many geometry problems.
What is a Tangent?
A tangent (think of a road that just grazes a round lake) meets the circle at a single point called the point of tangency. The magic fact is that the radius (the line from the centre to the edge) meets the tangent at a right angle (90°). Imagine a stick standing straight on a flat table; the stick is the radius, the table edge is the tangent.
Key properties of a tangent
- Touches the circle at exactly one point.
- The radius drawn to the point of tangency is perpendicular to the tangent.
- If two tangents are drawn from an external point, they are equal in length.
What is a Chord?
A chord is like a shortcut across a circular park: it joins any two points on the circle. If the chord passes through the centre, it becomes a special chord called a diameter, which is the longest possible chord.
Important chord facts
- The perpendicular bisector (a line that cuts it in half at 90°) of a chord passes through the centre.
- Equal chords are equally distant from the centre.
- The longer the chord, the closer it is to the centre.
Worked Example 1: Length of a tangent from an external point
Problem: From point P, 13 cm away from the centre O of a circle with radius 5 cm, find the length of the tangent PT.
Solution steps:
- Draw OP and the tangent PT. Connect OT (radius to point of tangency T).
- Because OT ⟂ PT, triangle O P T is a right‑angled triangle with OT = 5 cm and OP = 13 cm.
- Apply Pythagoras theorem (a² + b² = c²) where PT is the side we want:
- PT² = OP² – OT² = 13² – 5² = 169 – 25 = 144.
- Take the square root: PT = 12 cm.
So the tangent from P is 12 cm long.
Worked Example 2: Finding the distance of a chord from the centre
Problem: A circle has radius 10 cm. A chord AB is 16 cm long. How far is AB from the centre O?
Solution:
- Draw the radius OC that meets AB at its midpoint M (OC ⟂ AB).
- Now triangle O M A is right‑angled with OA = 10 cm (radius) and AM = half of AB = 8 cm.
- Use Pythagoras: OM² = OA² – AM² = 10² – 8² = 100 – 64 = 36.
- Thus OM = 6 cm. The chord is 6 cm away from the centre.
Quick Comparison Table
| Feature | Tangent | Chord |
|---|---|---|
| Touches the circle | Exactly one point | Two points |
| Relation with radius | Perpendicular at point of tangency | Perpendicular bisector passes through centre |
| Equal from external point | Two tangents from same external point are equal | Equal chords are equally distant from centre |
| Special case | None | Diameter (passes through centre) |
How to Find the Length of a Tangent – A Simple Flowchart
Common Mistakes to Avoid
- Assuming the radius is always inside the triangle formed by two tangents – it’s only perpendicular at the point of tangency.
- Mixing up the distance from the centre to a chord with the length of the chord itself.
- Forgetting that the perpendicular from the centre to a chord bisects it.
📝 Likely Exam Questions
- Question: Two tangents drawn from point P to a circle touch the circle at A and B. If PA = 7 cm, find PB.
- Answer: Tangents from the same external point are equal, so PB = 7 cm.
- Question: In a circle of radius 9 cm, a chord is 12 cm long. Find its distance from the centre.
- Answer: Half‑chord = 6 cm. Using Pythagoras: distance = √(9² – 6²) = √(81 – 36) = √45 = 3√5 cm.
- Question: From a point 15 cm from the centre of a circle of radius 8 cm, a tangent is drawn. Find the length of the tangent.
- Answer: Tangent² = 15² – 8² = 225 – 64 = 161, so tangent = √161 cm (≈12.69 cm).
- Question: Prove that the angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.
- Answer: Use the fact that the angle subtended by the chord at the centre is twice the angle at the circumference and that the radius is perpendicular to the tangent; then apply the cyclic quadrilateral theorem.