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In the figure (i) given below, AD is a diameter of a circle with center O. If AB || CD, prove that AB = CD.

In figure, AD is a diameter of a circle with center O. If AB || CD, prove that AB = CD. Circle, ML Aggarwal Understanding Mathematics Solutions ICSE Class 9.

Circles

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Answer

Draw OM ⊥ AB and ON ⊥ CD,

In figure, AD is a diameter of a circle with center O. If AB || CD, prove that AB = CD. Circle, ML Aggarwal Understanding Mathematics Solutions ICSE Class 9.

In △OAM and △ODN,

OA = OD (Radius of circle)

∠AOM = ∠DON (Vertically opposite angles are equal)

∠OMA = ∠OND (Both equal to 90°)

∴ △OAM ≅ △ODN (By A.S.A. axiom).

∴ ND = AM (By C.P.C.T.) ……..(1)

Since, the perpendicular to a chord from the centre of the circle bisects the chord,

so, equation 1 can be written as,

CD2=AB2\dfrac{\text{CD}}{2} = \dfrac{\text{AB}}{2}

⇒ AB = CD.

Hence, proved that AB = CD.

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