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    class 9 maths ex 10.4 question  2
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    Nitish Sharma
    Q.2      If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.
    Sol.
    Given : AB and CD are chords of a circle with centre O. AB and CD intersect at P and AB = CD.
    To prove : (i) AP = PD (ii) PB = CP.
    Construction : Draw OM
    ⊥
    AB , ON
    ⊥
    CD.
    Join OP,

    AM = MB =
    1
    2
    AB [Perpendicular from centre bisects the chord]
    CN = ND =
    1
    2
    CD [Perpendicular from centre bisects the chord]
    AM = ND and MB = CN [Since AB = CD (given)]
    In
    Δs
    OMP and ONP , we have
    OM = ON                                                [Equal chords of a circle are equidistant from the centre]
    ∠OMP=∠ONP
    [Since Each = 90º]
    OP = OP [Common]
    By RHS' criterion of congruence,
    ΔOMP≅ΔONP

    ⇒
    MP = PN ... (2) [C.P.C.T.]
    Adding (1) and (2) , we have
    AM MP = ND PN
    ⇒
    AP = PD
    Subtracting (2) from (1), we have
    MB – MP = CN – PN
    ⇒
    PB = CP
    Hence (i) AP = PD and (ii) PB = CP

    Written on Jan. 9, 2018, 11:14 p.m.

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