Circles: Chords, Angles, Tangents and Cyclic Quadrilaterals
Circles: Chords, Angles, Tangents and Cyclic Quadrilaterals
Circle geometry has a handful of theorems, and NTPC uses all of them: the perpendicular from the centre bisects a chord, the angle at the centre is twice the angle on the circle, tangents from a point are equal, the tangent-chord angle equals the angle in the alternate segment, and opposite angles of a cyclic quadrilateral add to 180°.
1. Theorems Box
| Theorem | Statement |
|---|---|
| Perpendicular from centre | Bisects the chord: (half chord)² + distance² = r² |
| Equal chords | Equidistant from the centre and subtend equal angles there |
| Angle at centre | Twice the angle at the circumference on the same arc |
| Same segment | Angles on the same arc are equal |
| Semicircle | Angle in a semicircle = 90° |
| Tangent and radius | Perpendicular at the point of contact |
| Two tangents from P | Equal lengths; ∠APB + ∠AOB = 180° |
| Alternate segment | Tangent-chord angle = angle in the alternate segment |
| Cyclic quadrilateral | Opposite angles add to 180°; an exterior angle = the interior opposite angle |
Tangent length from a point at distance d from the centre: √(d² − r²)
Direct common tangent of two circles (centres d apart): √(d² − (r₁ − r₂)²); transverse: √(d² − (r₁ + r₂)²)
✗ Arc AB subtends 110° at the centre, so 110° at the circumference | ✓ Half of it: 55°
✗ Two tangents from P make ∠APB = 70°, so ∠AOB = 70° | ✓ The quadrilateral PAOB has two right angles: ∠AOB = 110°
हिंदी नोट: केंद्र से जीवा पर लंब जीवा को समद्विभाजित करता है। केंद्र पर बना कोण परिधि पर बने कोण का दोगुना होता है। स्पर्शरेखा त्रिज्या पर लंब होती है। चक्रीय चतुर्भुज के सम्मुख कोणों का योग 180° होता है।
Exam Pointer: Verified NTPC patterns: a chord of concentric circles (July 2026 Graduate CBT-2), two tangents with an inscribed angle (March 2026 Graduate CBT-1), the alternate segment theorem (March 2026 Graduate CBT-1) and cyclic quadrilateral angles (May 2022 CBT-2 and March 2026 Graduate CBT-1). Angle-at-centre questions had no separate verified shift and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Chord, distance from the centre and concentric circles
[PYQ: NTPC Graduate CBT-2 10-Jul-2026]
EXAM LEVEL
Q. A chord of a circle of radius 13 cm is 24 cm long. Find its distance from the centre.
Half the chord is 12 cm, so the distance = √(169 − 144) = 5 cm.
Answer: 5 cm
EXAMATLAS LEVEL
Q. Two concentric circles have radii 13 cm and 5 cm. A chord of the larger circle touches the smaller circle. Find its length. In another circle of radius 13 cm, parallel chords of 10 cm and 24 cm lie on opposite sides of the centre; find the distance between them.
The chord is at distance 5 cm (the smaller radius), so half of it is √(169 − 25) = 12 cm and it is 24 cm long. For the parallel chords, the distances are √(169 − 25) = 12 cm and √(169 − 144) = 5 cm, so they are 17 cm apart.
Answer: 24 cm; 17 cm
Pattern 2: Angle at the centre, same segment and semicircle
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. An arc subtends 110° at the centre. What angle does it subtend at a point on the remaining part of the circle?
Half of 110°, that is 55°.
Answer: 55°
EXAMATLAS LEVEL
Q. AB is a diameter of a circle and C is a point on it with ∠CAB = 35°. D lies on the arc opposite to C, on the other side of AB. Find ∠ABC and ∠ADC.
∠ACB = 90° (semicircle), so ∠ABC = 55°. ∠ADC stands on arc AC, the same arc as ∠ABC, so ∠ADC = 55°.
Answer: 55° and 55°
Pattern 3: Tangents from an external point
[PYQ: NTPC Graduate CBT-1 17-Mar-2026 Shift-2]
EXAM LEVEL
Q. A point is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent from it.
√(169 − 25) = 12 cm.
Answer: 12 cm
EXAMATLAS LEVEL
Q. PA and PB are tangents to a circle with centre O, and ∠APB = 70°. C is a point on the major arc AB. Find ∠AOB, ∠ACB and ∠OAB.
In quadrilateral PAOB the angles at A and B are 90°, so ∠AOB = 180° − 70° = 110°. ∠ACB = 110°/2 = 55°. Triangle OAB is isosceles, so ∠OAB = (180° − 110°)/2 = 35°.
Answer: 110°, 55°, 35°
Pattern 4: Alternate segment theorem
[PYQ: NTPC Graduate CBT-1 18-Mar-2026 Shift-1]
EXAM LEVEL
Q. The tangent at P to a circle makes an angle of 50° with the chord PQ. Find the angle subtended by PQ at a point R on the circle in the alternate segment.
By the alternate segment theorem, ∠PRQ = 50°.
Answer: 50°
EXAMATLAS LEVEL
Q. The tangent at A makes an angle of 65° with the chord AB, and C lies on the major arc AB. Find ∠ACB, ∠AOB and ∠OAB, where O is the centre.
∠ACB = 65° (alternate segment). ∠AOB = 2 × 65° = 130°. ∠OAB = 90° − 65° = 25° (since OA ⊥ the tangent), which also equals (180° − 130°)/2.
Answer: 65°, 130°, 25°
Pattern 5: Cyclic quadrilateral
[PYQ: NTPC CBT-2 9-May-2022 Shift-2 | NTPC Graduate CBT-1 17-Mar-2026 Shift-2]
EXAM LEVEL
Q. In cyclic quadrilateral ABCD, ∠A = 75°. Find ∠C and the exterior angle at C formed by extending BC.
∠C = 180° − 75° = 105°. The exterior angle at C equals the interior opposite angle ∠A = 75°.
Answer: 105° and 75°
EXAMATLAS LEVEL
Q. In cyclic quadrilateral ABCD, ∠A : ∠C = 4 : 5 and ∠B is 30° less than twice ∠D. Find all four angles.
∠A + ∠C = 180° in the ratio 4 : 5, so ∠A = 80° and ∠C = 100°. ∠B + ∠D = 180° with ∠B = 2∠D − 30°, so 3∠D = 210°, ∠D = 70° and ∠B = 110°.
Answer: 80°, 110°, 100°, 70°
CBT-2 LEVEL
Pattern 6: Common tangents of two circles
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Two circles of radii 8 cm and 3 cm have centres 13 cm apart. Find the length of their direct common tangent.
√(169 − 25) = 12 cm.
Answer: 12 cm
EXAMATLAS LEVEL
Q. Two circles of radii 11 cm and 4 cm have centres 25 cm apart. Find the lengths of the direct and transverse common tangents.
Direct = √(625 − 49) = 24 cm. Transverse = √(625 − 225) = 20 cm.
Answer: 24 cm and 20 cm
60-Second Revision
- Half chord² + distance² = r².
- Centre angle = 2 × circumference angle; semicircle angle = 90°.
- Two tangents: ∠APB + ∠AOB = 180°; tangent length √(d² − r²).
- Tangent-chord angle = angle in the alternate segment.
- Cyclic quadrilateral: opposite angles add to 180°.