Problem
GEO-B1-M05-P010 Radius to the Midpoint of a Chord
#10
★★☆☆☆ Level 2 of 5
In a circle with centre \(O\), point \(M\) is the midpoint of chord \(AB\). Prove that \(OM\perp AB\).
Compare triangles \(OMA\) and \(OMB\).
We have \(OA=OB\), since these are radii, \(AM=MB\) by condition, and \(OM\) is common. Hence \(\triangle OMA\cong\triangle OMB\). Therefore \(\angle OMA=\angle OMB\). These angles are supplementary, so each is \(90^\circ\). Thus \(OM\perp AB\).
A useful fact for problems with chords and a centre.