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Circles

Short Answer Type

41.

In figure, AB and CD are equal chords of a circle whose centre is O. If OM ⊥ AB and ON ⊥ CD, prove that ∠OMN = ∠ONM.

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42.
AB and CD are equal chords of a circle whose centre is O. When produced, these chords meet at E. Prove that EB = ED and AE = CE.

Given: AB and CD are equal chords of a circle whose centre is O. When produced, these chords meet at E.
To Prove: EB = ED and AE = CE.

Given: AB and CD are equal chords of a circle whose centre is O. When

Construction: From O draw OP ⊥ AB and OQ ⊥ CD. Join OE.
Proof: ∵ AB = CD    | Given
∴ OP = OQ
| ∵ Equal chords of a circle are equidistant from the centre
Now in right ∆s OPE and OQE,
Hyp. OE = Hyp. OE    | Common
Side OP = Side OQ    | Proved above

$therefore space space space space increment OPE space equals space increment AQU space space space space space space space space space space space space space space space space space space left enclose space straight R. straight H. straight S. space Axiom end enclose therefore space space space space space PE equals QE space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left enclose space straight C. straight P. straight C. straight T. end enclose rightwards double arrow space space space space space PE minus 1 half AB equals QE minus 1 half CD space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left enclose space because space AB space equals space CD space left parenthesis Given right parenthesis end enclose$

$rightwards double arrow$ PE - PB = QE - QD
$rightwards double arrow$ EB = ED Proved.
$rightwards double arrow$ BE + AB = ED + CD    $left enclose space space space because space space AB equals CD end enclose$
$rightwards double arrow$ AE = CE

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43. Bisector AD of ∠BAC of ∆ABC passes through the centre O of the circumcircle of ∆ABC. Prove that AB = AC.

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44. In figure, AB and AC are two equal chords of a circle whose centre is O. If OD ⊥ AB and OE ⊥ AC, prove that ADE is an isosceles triangle.

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45. If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, prove that the chords are equal.

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46. In figure, AB and CD are two chords of a circle with centre O. The two chords intersect each other at the point P. If PO bisects ∠APD, prove that AB = CD.

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47. In the given figure, two chords AB and CD of a circle, with centre O, are such that OP and OQ, the perpendiculars from the centre to the chords, are equal. Show that AB = CD.

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48. Prove that the line segment joining the mid-points of two equal chords of a circle makes equal angles on its same side with the chords.

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49. Prove that the chords equidistant from the centre of a circle are equal in length.

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50. PQ and RQ are chords of a circle equidistant from the centre. Prove that the diameter passing through Q is the bisector of ∠PQR.

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Previous Year Papers

Test Series

Test Yourself

Short Answer Type

42. AB and CD are equal chords of a circle whose centre is O. When produced, these chords meet at E. Prove that EB = ED and AE = CE.

42.
AB and CD are equal chords of a circle whose centre is O. When produced, these chords meet at E. Prove that EB = ED and AE = CE.