Circles — Chords & Tangents
Circle — chord और tangent के rules
Circles — Chords & Tangents
- Geometry
- Circles — Chords & Tangents
Use chord, tangent and semicircle properties to solve length and angle questions.
🎯 Learning Objective
Use chord, tangent and semicircle properties to solve length and angle questions.
💡 Concept
- Radius r, diameter = 2r — the diameter is the longest chord
- Perpendicular from the centre to a chord bisects the chord
- Half-chord, distance from centre and radius form a right triangle
- Tangent ⊥ radius at the point of contact
- Angle in a semicircle is always 90°
🧮 Key Formulas
(half chord)² + (distance from centre)² = r²
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Tangent² = OP² − r²
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Angle in semicircle = 90°
✏️ Easy Example
Q. A chord is 3 cm from the centre of a circle of radius 5 cm. Find the length of the chord.
- Half chord = √(5² − 3²)
- = √16 = 4
- Chord = 2 × 4
Answer: 8 cm
🇮🇳 Real-Life Example
A bicycle wheel: every spoke is a radius, and the road it rests on is a tangent — always perpendicular to the spoke touching the ground.
📝 Exam-Level Example
Q. AB is the diameter of a circle and C is a point on it. If ∠CAB = 35°, find ∠CBA.
- Angle in semicircle: ∠ACB = 90°
- ∠CBA = 180 − 90 − 35
Answer: 55°
📝 Exam-Level Example
Q. From a point P, 10 cm from the centre O of a circle of radius 6 cm, find the length of the tangent PA.
- ∠OAP = 90° → right triangle
- PA² = 10² − 6² = 64
- PA = √64
Answer: 8 cm
🪄 Memory Trick
Every chord/tangent length question is one hidden right triangle — and it is usually a 3-4-5 or 6-8-10 in disguise.
⚠️ Common Mistakes
- ❌ Using the full chord instead of half chord in the right triangle
- ❌ Forgetting the 90° between tangent and radius at the contact point
🏆 Exam Tips
- ✅ Diameter in the figure → hunt for the hidden 90° in the semicircle
- ✅ Equal chords are equidistant from the centre — use it for symmetry questions
📌 Summary
- Perpendicular from centre bisects the chord
- (half chord)² + d² = r²
- Tangent ⊥ radius; tangent² = OP² − r²
- Semicircle angle = 90°