Let ABCD be a cyclic quadrilateral where diagonal AC and BD are diameters.
Since BD is a diameter.
Arc BAD is a semicircle, So ∠BAD = 90° (Angle in a semi circle is a right angle)
Since AC ia a diameter.
Arc ABC is a semicircle, So ∠ABC = 90° (Angle in a semi circle is a right angle)
Also,
ABCD is a cyclic quadrilateral
From figure,
⇒ ∠BCD + ∠BAD = 180° (Sum of opposite angles of cyclic quadrilateral is 180°)
⇒ ∠BCD + 90° = 180°
⇒ ∠BCD = 180° - 90°
⇒ ∠BCD = 90°.
⇒ ∠ABC + ∠ADC =180° (Sum of opposite angles of cyclic quadrilateral is 180°)
⇒ ∠ADC + 90° = 180°
⇒ ∠ADC = 180° - 90°
⇒ ∠ADC = 90°.
So, in quadrilateral ABCD
∠A = ∠B = ∠C = ∠D = 90°
Since, all angles equal to 90°.
Hence, proved that ABCD is a rectangle.
BRIGHT TUTORIALS
BRIGHT TUTORIALS
CBSE Class IX | Academic Year 2026-2027
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Mathematics | CirclesWeb Content • Interactive Notes
Circles — Interactive Study Guide
Circle Theorems Quick Reference
- Equal chords ⇔ equal central angles
- Perpendicular from centre bisects the chord
- Equal chords ⇔ equidistant from centre
- Central angle = 2 × inscribed angle (same arc)
- Angles in same segment are equal
- Angle in semicircle = 90°
- Opposite angles of cyclic quad = 180°
Problem-Solving Toolkit
For chord problems: Drop perpendicular from centre → bisects chord → use Pythagoras.
For angle problems: Identify if angle is at centre or circumference → apply the 2x rule.
For cyclic quad: Opposite angles add up to 180°.
Quick Self-Check
- Inscribed angle = 35°. Central angle for the same arc? (70°)
- Chord = 10 cm, distance from centre = 12 cm. Radius? (√(25+144) = √169 = 13 cm)
- ABCD is cyclic, ∠A = 95°. Find ∠C. (85°)