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CBSE Class 9 Mathematics Question 3 of 12

Circles — Question 4

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Question 4

If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD.

If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD. NCERT Class 9 Mathematics CBSE Solutions.
Answer

Draw a perpendicular from the center of the circle to the line AD, such that OM ⊥ AD

BC and AD are the chords of smaller circle and the bigger circle respectively.

If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD. NCERT Class 9 Mathematics CBSE Solutions.

We know that,

Perpendicular from the center bisects the chord.

⇒ BM = MC ......... (1)

⇒ AM = MD ......... (2)

By subtracting equation (1) from (2), we get :

⇒ AM − BM = MD − MC

⇒ AB = CD

Hence, proved that AB = CD.

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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