Class IX Math NCERT Solution for Circles 10.2

EXERCISE 10.2 (Page 173)
Q1.  Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent cirlces subtend equal angles at their centres.
Sol: We have a circle having its centre at O and two equal chords AB and CD such that they subtend ∠AOB and ∠COD respectively at the centre, i.e. at O.
          We have to prove that
               ∠AOB = ∠COD
          Now, in ΔAOB and ΔCOD, we have
               AO = CO                                                  [Radii of the same circle]
               BO = DO                                                  [Radii of the same circle]
               AB = CD                                                   [Given]
          ∴ ΔAOB &becong; ΔCOD                      [SAS criterion]
          ∴Their correspondong parts are equal
          ∴ ∠AOB = ∠COD
Q2.  Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
Sol: We have a circle having its centre at O, and its two chords AB and CD such that
                 ∠AOB = ∠COD
          We have to prove that
               AB = CD
          ∵In ΔAOB and ΔCOD, we have:
               AO = CO                                             [Radii of the same circle]
               BO = DO                                             [Radii of the same circle]
               ∠AOB = ∠COD                                [Given]
          ∴ ΔAOB ≌ ΔCOD                                 [SAS criterion]
          ∴Their correspondong parts are equal, i.e. AB - CD

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