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CBSE Class 9 Mathematics Question 12 of 12

Circles — Question 12

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Question 12

Prove that a cyclic parallelogram is a rectangle.

Answer

Let ABCD be the cyclic parallelogram.

Prove that a cyclic parallelogram is a rectangle. NCERT Class 9 Mathematics CBSE Solutions.

We know that opposite angles of a parallelogram are equal.

⇒ ∠A = ∠C .......(1)

⇒ ∠B = ∠D .......(2)

We know that the sum of opposite angles of a cyclic quadrilateral is 180°.

⇒ ∠A + ∠C = 180°

⇒ ∠A + ∠A = 180° (From equation (1))

⇒ 2∠A = 180°

⇒ ∠A = 180°2\dfrac{180°}{2}

⇒ ∠A = 90°.

We know that,

If one angle of a parallelogram is 90°, then it is a rectangle.

Thus, quadrilateral ABCD is a rectangle.

Hence, proved that a cyclic parallelogram is a rectangle.

Circles - Interactive Study Notes | Bright Tutorials
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Mathematics | CirclesWeb Content • Interactive Notes

Circles — Interactive Study Guide

Circle Theorems Quick Reference

  1. Equal chords ⇔ equal central angles
  2. Perpendicular from centre bisects the chord
  3. Equal chords ⇔ equidistant from centre
  4. Central angle = 2 × inscribed angle (same arc)
  5. Angles in same segment are equal
  6. Angle in semicircle = 90°
  7. 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

  1. Inscribed angle = 35°. Central angle for the same arc? (70°)
  2. Chord = 10 cm, distance from centre = 12 cm. Radius? (√(25+144) = √169 = 13 cm)
  3. ABCD is cyclic, ∠A = 95°. Find ∠C. (85°)

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