M and N are the mid-points of two equal chords AB and CD respectively of a circle with center O. Prove that :
(i) ∠BMN = ∠DNM
(ii) ∠AMN = ∠CNM.

Question

M and N are the mid-points of two equal chords AB and CD respectively of a circle with center O. Prove that :  (i) ∠BMN = ∠DNM (ii) ∠AMN = ∠CNM.

KnowledgeBoat · Accepted Answer

Join OM, ON, OB and OD. Given, M and N are the mid-points of two equal chords AB and CD. ∴ BM = and DN = Since, AB and CD are equal chords. ∴ BM = DN ..........(1) We know that, A straight line drawn from the center of a circle to bisect a chord, which is not a diameter, is at right angles to the chord. ∴ OM ⊥ AB and ON ⊥ CD. ∴ ∠OMB = ∠OND (Both equal to 90°) ........(2) (i) In triangle OMN, ⇒ ON = OM (Equal chords of a circle are equidistant from the center.) ⇒ ∠OMN = ∠ONM (Angles opposite to equal sides are equal) ........(3) Subtracting equation (2) from (3), we get : ⇒ ∠OMB - ∠OMN = ∠OND - ∠ONM ⇒ ∠BMN = ∠DNM. Hence, proved that ∠BMN = ∠DNM. (ii) From figure, ⇒ ∠OMA = ∠ONC (Both equal to 90°) ...............(4) Adding equation (3) and (4), we get : ⇒ ∠OMA + ∠OMN = ∠ONC + ∠ONM ⇒ ∠AMN = ∠CNM. Hence, proved that ∠AMN = ∠CNM.

Mathematics

M and N are the mid-points of two equal chords AB and CD respectively of a circle with center O. Prove that :
(i) ∠BMN = ∠DNM
(ii) ∠AMN = ∠CNM.

Circles

Related Questions

A chord CD of a circle, whose center is O, is bisected at P by a diameter AB.
Given OA = OB = 15 cm and OP = 9 cm. Calculate the lengths of :
(i) CD
(ii) AD
(iii) CB.

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Prove that the chords AB and CD, which are intercepted by the two circles, are equal.

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Mathematics

M and N are the mid-points of two equal chords AB and CD respectively of a circle with center O. Prove that :(i) ∠BMN = ∠DNM(ii) ∠AMN = ∠CNM.

Circles

Related Questions

A chord CD of a circle, whose center is O, is bisected at P by a diameter AB.Given OA = OB = 15 cm and OP = 9 cm. Calculate the lengths of :(i) CD(ii) AD(iii) CB.

A straight line is drawn cutting two equal circles and passing through the mid-point M of the line joining their centers O and O'.Prove that the chords AB and CD, which are intercepted by the two circles, are equal.

Two equal chords AB and CD of a circle with center O, intersect each other at point P inside the circle. Prove that :(i) AP = CP(ii) BP = DP

In the following figure, OABC is a square. A circle is drawn with O as center which meets OC at P and OA at Q. Prove that :(i) △ OPA ≅ △ OQC(ii) △ BPC ≅ △ BQA

M and N are the mid-points of two equal chords AB and CD respectively of a circle with center O. Prove that :
(i) ∠BMN = ∠DNM
(ii) ∠AMN = ∠CNM.

A chord CD of a circle, whose center is O, is bisected at P by a diameter AB.
Given OA = OB = 15 cm and OP = 9 cm. Calculate the lengths of :
(i) CD
(ii) AD
(iii) CB.

A straight line is drawn cutting two equal circles and passing through the mid-point M of the line joining their centers O and O'.
Prove that the chords AB and CD, which are intercepted by the two circles, are equal.

Two equal chords AB and CD of a circle with center O, intersect each other at point P inside the circle. Prove that :
(i) AP = CP
(ii) BP = DP

In the following figure, OABC is a square. A circle is drawn with O as center which meets OC at P and OA at Q. Prove that :
(i) △ OPA ≅ △ OQC
(ii) △ BPC ≅ △ BQA