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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. Q.2

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How i solve the question of class 9th ncert math of Circles chapter of exercise 10.1of question no (2). I think it is very important question of class 9th give me the tricky way for solving this question 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.

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  1. This answer was edited.

    Let AB and CD be two equal cords (i.e. AB = CD). In the above question, it is given that AB and CD intersect at a point, say, E.

    It is now to be proven that the line segments AE = DE and CE = BE

    Construction Steps:

    Step 1: From the center of the circle, draw a perpendicular to AB i.e. OM ⊥ AB

    Step 2: Similarly, draw ON ⊥ CD.

    Step 3: Join OE.

    Now, the diagram is as follows-

    Ncert solutions class 9 chapter 10-11

    Proof:

    From the diagram, it is seen that OM bisects AB and so, OM ⊥ AB

    Similarly, ON bisects CD and so, ON ⊥ CD

    It is known that AB = CD. So,

    AM = ND — (i)

    and MB = CN — (ii)

    Now, triangles ΔOME and ΔONE are similar by RHS congruency since

    OME = ONE (They are perpendiculars)

    OE = OE (It is the common side)

    OM = ON (AB and CD are equal and so, they are equidistant from the centre)

    ∴ ΔOME ΔONE

    ME = EN (by CPCT) — (iii)

    Now, from equations (i) and (ii) we get,

    AM+ME = ND+EN

    So, AE = ED

    Now from equations (ii) and (iii) we get,

    MB-ME = CN-EN

    So, EB = CE (Hence proved).

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