Why Tangents and Chords Matter in Circle Problems?
Ever wondered why a single line can just kiss a circle at one spot, or how a straight line inside a circle can split it into two perfect arcs? Those are the magic of tangents and chords – the shortcuts that make many ICSE circle questions solvable in a flash.
In simple words, a tangent is a line that touches a circle at exactly one point, while a chord is a straight line that joins any two points on the circle. Both have special relationships with the circle’s centre that let you find missing lengths without heavy algebra.
What is a Tangent?
A tangent (think of a road just grazing a round pond) is a straight line that meets the circle at only one point, called the point of contact.
Key property of a tangent
The most useful fact is that the radius drawn to the point of contact is perpendicular (forms a 90° angle) to the tangent. Imagine the radius as a stick standing straight up from the ground; the tangent is a flat board leaning against it – they meet at a right angle.
What is a Chord?
A chord is any line segment whose ends both lie on the circle. If you draw a string across a circular pizza, that string is a chord.
Important chord facts
- Perpendicular bisector rule: A line drawn from the centre of the circle to the midpoint of a chord is perpendicular to the chord and also bisects (splits) it into two equal halves.
- Equal chords, equal distances: If two chords are the same length, they are equally far from the centre. Conversely, chords that are the same distance from the centre have the same length.
Tangent‑Chord Theorem (Alternate Segment Theorem)
This theorem says that the angle between a tangent and a chord through the point of contact equals the angle in the alternate segment (the angle formed by the chord with another point on the circle opposite the tangent).
How to use the theorem
1. Identify the tangent and the chord sharing the point of contact.
2. Locate the “alternate segment” – the part of the circle opposite the tangent.
3. Set the angle between the tangent and chord equal to the angle subtended by the chord in that opposite segment.
Worked Example 1: Finding the Length of a Tangent
Problem: In a circle with centre O, the radius is 7 cm. A point P lies outside the circle such that OP = 13 cm. Find the length of the tangent PT drawn from P to the circle.
Solution: Connect O to T (the point of contact). OT = radius = 7 cm. OP = 13 cm. Triangle OTP is right‑angled at T (because radius OT ⟂ tangent PT). Using the Pythagorean theorem (a² + b² = c²):
- OT² + PT² = OP²
- 7² + PT² = 13²
- 49 + PT² = 169
- PT² = 120 → PT = √120 ≈ 10.95 cm
Worked Example 2: Using Chord Properties to Find a Missing Segment
Problem: In a circle of radius 10 cm, a chord AB is 12 cm long. Find the distance from the centre O to the chord (i.e., the length of the perpendicular OM where M is the midpoint of AB).
Solution: Draw OM ⟂ AB, with M the midpoint of AB. Because OM bisects AB, AM = BM = 6 cm. Triangle OMA is right‑angled at M.
- OA = radius = 10 cm
- AM = 6 cm
- Using Pythagoras: OM² + AM² = OA²
- OM² + 6² = 10² → OM² + 36 = 100 → OM² = 64 → OM = 8 cm
Quick Comparison: Tangent vs Chord
| Feature | Tangent | Chord |
|---|---|---|
| Definition | Line touching circle at exactly one point | Segment joining two points on the circle |
| Key right‑angle property | Radius to point of contact ⟂ tangent | Line from centre to midpoint ⟂ chord |
| Length formula | Using right triangle: PT = √(OP² – r²) | Using right triangle: distance from centre = √(r² – (½ chord)²) |
| Angle relation | Tangent‑chord theorem links angle with opposite arc | Equal chords ↔ equal distances from centre |
📝 Likely Exam Questions
- In a circle of radius 5 cm, a tangent from an external point P is 12 cm long. Find the distance OP.
Answer: OP² = PT² + r² = 12² + 5² = 144 + 25 = 169 → OP = 13 cm. - Two chords of a circle are each 8 cm long and are 6 cm from the centre. Prove that the chords are equal and find the radius of the circle.
Answer: Using OM² + (½ chord)² = r² → 6² + 4² = r² → r² = 36 + 16 = 52 → r = √52 cm. - Given a circle with centre O, a tangent at point T makes a 30° angle with chord TA. Find the angle in the alternate segment (angle TBA).
Answer: By tangent‑chord theorem, ∠(tangent, chord) = ∠TBA → ∠TBA = 30°. - In a circle, the distance from the centre to a chord is half the radius. If the radius is 14 cm, find the length of the chord.
Answer: Distance = 7 cm. Use right triangle: (½ chord)² = r² – d² = 14² – 7² = 196 – 49 = 147 → half‑chord = √147 ≈ 12.12 cm → chord ≈ 24.24 cm. - Show that the tangent drawn from an external point to a circle is equal in length to the other tangent from the same point.
Answer: Triangles formed by the two tangents and the radii are congruent (RHS), giving PT = PT'.