Why tangents and chords are worth a second look
Imagine a bike wheel rolling on a flat road. The point where the tyre kisses the ground is a perfect example of a tangent – it touches the circle at just one spot. Knowing how that tiny touch works can unlock many circle problems you’ll face in ICSE exams.
In Simple Words: A tangent is a straight line that just grazes a circle, while a chord is a straight line that cuts straight through the circle, joining two points on its edge. Both follow neat rules that let you calculate lengths and angles quickly.
What exactly is a tangent?
A tangent (think of a pencil just barely touching a ball) is a line that meets a circle at one single point. The magic rule is: the radius drawn to that point of contact is always perpendicular (forms a 90° angle) to the tangent.
Key tangent properties
- It touches the circle at only one point.
- The radius to the point of contact is at right angles to the tangent.
- If two tangents are drawn from the same external point, they are equal in length.
- The square of a tangent’s length equals the product of the whole and external segments of any secant through the same external point (Power of a Point).
What is a chord?
A chord is simply a line that joins any two points on the circle’s boundary. Picture a piece of string stretched across a pizza – that’s a chord.
Important chord properties
- The perpendicular drawn from the centre to a chord bisects (splits) the chord.
- All chords that are equal in length are equally distant from the centre.
- When two chords intersect inside the circle, the product of the segments of one chord equals the product of the segments of the other (Intersecting Chords Theorem).
Connecting tangents and chords: the theorems you’ll use
Tangent‑Chord Theorem: The angle formed between a tangent and a chord through the point of contact equals the angle in the alternate segment (the angle subtended by the chord at any point on the opposite arc).
Power of a Point: For any point outside the circle, the square of the tangent length equals the product of the whole secant length and its external part. In symbols, if PT is a tangent and PAB is a secant, then PT² = PA·PB.
Worked example 1 – finding a tangent length
Given: A circle has centre O and radius 6 cm. An external point P is 10 cm from O. Find the length of the tangent PT drawn from P to the circle.
Solution steps:
- Draw OP, PT and the radius OT. Triangle OPT is right‑angled at T because OT ⟂ PT.
- Apply Pythagoras: OP² = OT² + PT².
- Plug in the numbers: 10² = 6² + PT² ⇒ 100 = 36 + PT².
- Subtract 36: PT² = 64.
- Take the square root: PT = 8 cm.
Worked example 2 – using the Tangent‑Chord Theorem
Given: In a circle, chord AB subtends an angle of 40° at the centre. A tangent at point A meets the extension of chord AB at point T. Find the angle ∠BAT.
Solution:
- The angle subtended by AB at the centre is 40°, so the angle in the opposite arc (the angle on the far side of the chord) is half of that: 20°.
- By the Tangent‑Chord Theorem, ∠BAT (tangent‑chord angle) equals the angle in the opposite segment, which is 20°.
Quick comparison table
| Feature | Tangent | Chord |
|---|---|---|
| Touches the circle | Exactly one point | Two points |
| Relation with radius | Perpendicular at point of contact | Perpendicular from centre bisects it |
| Length from external point | Equal for all tangents from same point | Varies; governed by intersecting‑chords rule |
| Key theorem | Power of a Point | Intersecting Chords Theorem |
Step‑by‑step to solve a typical tangent‑chord question
📝 Likely Exam Questions
- Question: From a point 13 cm away from the centre of a circle of radius 5 cm, two tangents are drawn. Find the length of each tangent.
Answer: Using PT² = OP² – OT² ⇒ PT = √(13² – 5²) = √(169 – 25) = √144 = 12 cm. - Question: In a circle, the chord CD is 8 cm long and is 3 cm from the centre. Find the radius.
Answer: Draw the perpendicular from the centre to the chord, forming a right triangle with half‑chord (4 cm) and distance 3 cm. Radius² = 4² + 3² ⇒ r = 5 cm. - Question: A tangent at point P makes a 30° angle with chord PQ. What is the angle subtended by PQ at the opposite arc?
Answer: By the Tangent‑Chord Theorem, the angle in the opposite segment equals 30°. - Question: Two secants PAB and PCD intersect outside a circle at P. PA = 6 cm, PB = 14 cm, PC = 4 cm. Find PD.
Answer: Power of a Point: PA·PB = PC·PD ⇒ 6·14 = 4·PD ⇒ 84 = 4·PD ⇒ PD = 21 cm. - Question: Prove that the perpendicular from the centre of a circle to any chord bisects the chord.
Answer: Draw radius OM to midpoint M of chord AB. Triangle OMA is right‑angled at M. By symmetry, OA = OB, so M must be the midpoint; otherwise the two right triangles would have different hypotenuses, contradicting OA = OB.