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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.

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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.

Given: In figure, AB and AC are two equal chords of a circle whose centre is O. OD ⊥ AB and OE ⊥ AC.
To Prove: ADE is an isosceles triangle.
Proof: ∵ AB = AC
∴ OD = OE
| ∵ Equal chords are equidistant from the centre

∴ In ∆ODE,
∠ODE = ∠OED
| Angle opposite to equal sides
⇒ 90° - ∠ODE = 90° - ∠OED
⇒ ∠ODA - ∠ODE = ∠OEA - ∠OED
⇒ ∠ADE = ∠AED
∴ AD = AE
| Sides opposite to equal angles
∴ ∆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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