Why do tangents and chords matter?
Ever watched a line just skim the edge of a circle and thought, "That looks neat!"? That line is a tangent, and the segment that cuts across the circle is a chord. Both show up in every ICSE geometry paper, and knowing their tricks can shave minutes off your answer time.
💡 In Simple Words: A tangent touches a circle at exactly one point, while a chord joins two points on the circle. The magic happens when you see how the centre, radius, and these lines interact – it’s like a secret handshake for circle problems.
What is a Tangent?
Tangent (first time): a straight line that meets a circle at just one point, called the point of contact. Think of a road that just brushes the edge of a round pond without driving into it.
Key fact: The radius (a line from the centre to the edge) is always perpendicular (forms a 90° angle) to the tangent at the point of contact. Imagine the radius as a pipe delivering water straight to the edge; the tangent is a board that leans against the pipe at a right angle.
Key Tangent Theorems
- Tangent‑Radius Perpendicularity: The radius drawn to the point where the tangent touches the circle meets the tangent at 90°.
- Tangent‑Chord Angle 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 side of the circle).
- Tangent‑Secant Theorem (also called the Power of a Point): If a tangent and a secant (a line that cuts the circle at two points) start from the same external point, then tangent length² = external part × whole secant.
Chord Basics
A chord is a line segment whose endpoints both lie on the circle. The longest chord is the diameter, which passes through the centre.
Important property: The line joining the centre to the midpoint of a chord is perpendicular to the chord. Picture the chord as a bridge across a river; the centre is a lighthouse that shines straight down to the middle of the bridge, forming a right angle.
Linking Tangents and Chords
When a tangent meets a chord at the point of contact, two useful relationships appear:
- The angle between the tangent and the chord equals the angle subtended by the chord in the opposite arc.
- If you drop a perpendicular from the centre to the chord, you can split the chord into two equal halves, which helps calculate lengths using the Pythagorean theorem.
These facts let you turn a seemingly messy picture into a set of right‑angled triangles you can solve with simple algebra.
Worked Example: Finding the Length of a Tangent from an External Point
Problem: A circle has centre O and radius 5 cm. A point P lies 13 cm from O. Find the length of the tangent PT that touches the circle at T.
Solution Steps:
- Draw OP, the line from the centre to the external point. Mark the radius OT (5 cm) and the unknown tangent PT.
- Recall the Tangent‑Radius Perpendicularity: OT ⟂ PT, so triangle OTP is right‑angled at T.
- Apply the Pythagorean theorem (a² + b² = c²) to triangle OTP, where OP is the hypotenuse (13 cm) and OT is one leg (5 cm).
- Compute PT:
PT² = OP² – OT² = 13² – 5² = 169 – 25 = 144. - Take the square root: PT = √144 = 12 cm.
So the tangent from P to the circle is 12 cm long.
Quick Summary Table
| Concept | Key Formula / Rule |
|---|---|
| Radius ⟂ Tangent | OT ⟂ PT at point of contact |
| Angle between Tangent & Chord | Equals angle in the opposite arc |
| Midpoint of Chord | Line from centre to midpoint is ⟂ chord |
| Tangent‑Secant (Power of a Point) | PT² = PA × PB (external × whole secant) |
| Length of Tangent from external point | PT = √(OP² – r²) |
📝 Likely Exam Questions
- Question: In a circle of radius 7 cm, a tangent from an external point P is 24 cm long. Find the distance OP.
Answer: OP² = PT² + r² = 24² + 7² = 576 + 49 = 625 ⇒ OP = 25 cm. - Question: A chord AB is 10 cm long and is 6 cm from the centre O. Find the radius of the circle.
Answer: Draw OM ⟂ AB, where M is midpoint of AB (5 cm). Using Pythagoras: r² = OM² + 5² ⇒ r² = 6² + 5² = 36 + 25 = 61 ⇒ r = √61 cm. - Question: Prove that the angle between a tangent and a chord equals the angle in the alternate segment.
Answer: Use the fact that the inscribed angle subtended by the chord equals the angle at the centre, and the tangent‑radius perpendicularity creates two congruent triangles, leading to equal angles. - Question: From a point P outside a circle, the lengths of two tangents drawn to the circle are equal. Explain why.
Answer: Tangents from a common external point are congruent because they share the same external point and each forms a right‑angled triangle with the radius; the hypotenuse OP is common, so the legs (tangents) must be equal. - Question: If a secant through P cuts the circle at A and B, and a tangent PT is drawn, show that PA·PB = PT².
Answer: Apply the Power of a Point theorem: product of external segment (PA) and whole secant (PB) equals square of the tangent length PT.