Why Circle Tangents and Chords Matter
Ever watched a bike wheel roll and wondered why the road touches the tyre at just one point? That tiny touch‑point is a real‑world example of a tangent – a line that kisses a circle without cutting through it. Mastering these ideas helps you ace geometry questions and even understand everyday curves.
💡 In Simple Words: A tangent is a straight line that meets a circle at exactly one point. A chord is any line segment whose ends both lie on the circle. Both have neat rules that let you find unknown lengths quickly.
Key Definitions (First Time Explained)
- Tangent: A line that touches a circle at one point only. Think of it as a gentle tap on the circle’s surface.
- Point of Tangency: The single spot where the tangent meets the circle.
- Radius: A line from the centre of the circle to any point on its edge.
- Chord: A straight line joining two points on the circle. If the chord passes through the centre, it becomes a diameter.
- Perpendicular: Two lines that meet at a right‑angle (90°).
Fundamental Properties
1. Tangent ⟂ Radius at the Point of Tangency
If you draw a radius to the point where a tangent touches the circle, the radius stands straight up like a wall against the tangent. This right‑angle rule is a go‑to for many problems.
2. Equal Tangents from an External Point
Pick any point outside the circle. If you draw two tangents from that point to the circle, both tangents have the same length. It’s like two equal‑length strings pulling from the same knot.
3. Angle Between a Tangent and a Chord
The angle formed by a tangent and a chord (at the point of contact) equals the angle in the alternate segment – the angle subtended by the same chord on the opposite side of the circle.
4. Chord Bisected by a Diameter or Perpendicular
If a diameter (a chord through the centre) cuts another chord, it does so at the chord’s midpoint. Also, a line drawn from the centre perpendicular to a chord bisects that chord.
Worked Example 1: Finding a Tangent Length
Problem: From point P, 13 cm away from the centre O, a tangent PT touches the circle at T. The radius OA is 5 cm. Find PT.
Solution Steps:
- Draw OP, PT and OA. OP is the line from centre to external point.
- Remember that OT (radius) is ⟂ PT, forming a right‑angled triangle O‑T‑P.
- Use Pythagoras:
OP² = OT² + PT²→13² = 5² + PT². - Calculate:
169 = 25 + PT²→PT² = 144→PT = 12 cm.
So the tangent from P is 12 cm long.
Worked Example 2: Chord Length Using Perpendicular Distance
Problem: A circle has radius 10 cm. A chord is 6 cm away from the centre. Find the chord’s length.
Solution Steps:
- Drop a perpendicular from the centre O to the chord; call the foot M. OM = 6 cm.
- Because OM bisects the chord, each half is a right‑angled triangle with hypotenuse OA (radius) and one leg OM.
- Apply Pythagoras:
OA² = OM² + AM²→10² = 6² + AM². - Calculate:
100 = 36 + AM²→AM² = 64→AM = 8 cm. - Full chord = 2 × AM = 16 cm.
Quick Comparison Table
| Feature | Tangent | Chord |
|---|---|---|
| Touches circle | Exactly one point | Two points (ends) |
| Relation with radius | Perpendicular at point of contact | Perpendicular from centre bisects only if drawn |
| From external point | Equal lengths for the two possible tangents | Not applicable |
| Forms right triangle | Yes, with radius and line to external point | Often part of right triangle when perpendicular drawn |
Problem‑Solving Flowchart
Tips to Remember for Exams
- Always draw a clear diagram – a picture saves time.
- Mark right angles whenever a radius meets a tangent.
- When two tangents start from the same external point, set their lengths equal.
- Use the “perpendicular bisects chord” rule to split a chord into two equal halves.
- Recall that the angle between a tangent and a chord equals the angle in the opposite segment.
📝 Likely Exam Questions
- Question: In a circle of radius 7 cm, a chord AB is 24 cm from the centre. Find the length of AB.
Answer: Drop OM = 24 cm, OA = 7 cm. Using Pythagoras, AM² = 7² − 24² is impossible, so the data is inconsistent – such a chord cannot exist. (Exam trick: check feasibility first.) - Question: From a point 15 cm from the centre of a circle with radius 9 cm, two tangents are drawn. Find the length of each tangent.
Answer: Right triangle O‑P‑T: PT² = OP² − OT² = 15² − 9² = 225 − 81 = 144 → PT = 12 cm. - Question: A chord CD is 10 cm long and is 6 cm from the centre. Find the radius of the circle.
Answer: Half‑chord = 5 cm. Using Pythagoras: r² = 6² + 5² = 36 + 25 = 61 → r = √61 cm ≈ 7.81 cm. - Question: Prove that the angle between a tangent and a chord equals the angle in the alternate segment.
Answer: Draw triangle with centre, use isosceles triangle properties and the fact that the angle at the centre subtended by the chord is twice the angle at the circumference (inscribed angle theorem). Then show the two angles are equal.