prove that line segment joining midpoints of two equal chords of a circle makes equal angles with the chord

Given that : AB and CD are two equal chords. And, M, N are mid point of chord AB and CD respectively.
To prove : AMN=CNM and BMN=DNM
Construction : Join OM and ON
Proof :
Since the line segment joining the centre of a circle to the midpoint of a chord is perpendicular to the chord. Therefore,
Since AB and CD are equal chords. So, they are equidistant from the other. (i.e., OM =ON)
In ΔOMN,OM=ONOMN=ONM   angles opposite to equal sides      .....1OMA=ONC   each 900      .....2And, OMB=OND   each 900      .....3Subtracting 1 and 2, we have,OMA-OMN=ONC-ONMAMN=CNMAnd,Adding 1 and 3, we have,OMB+OMN=OND+ONMBMN=DNM
Hence Proved.

  • 64

the line joining the mid point of a chord to center always forms an angle of 90o.

  • -13

please can someone answer this question its very important!!!!!!!!!!!!!!!!!!

  • 2
What are you looking for?