Question

The line joining the centre of a circle to the mid-point of a chord is perpendicular to the chord.​

Answer 1

yes the line joining the centre of a circle to the midpoint of a chord is perpendicular to chord and bisecting the chord into two equal parts and making an angle of 90° ........

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Answer 2

AnswEr:

Given : A chord PQ of a circle C(O,r) with the mid-point M.

To prove : Om $\perp$ PQ

Proof : In triangles OPM and OQM, we have

OP = OQ [Radii of the same circle ]

PM = MQ [ M is the mid-point of PQ]

OM = OM

So, by SSS criterion of congruence, we have

$\qquad \triangle \sf \: OPM \cong \triangle \: OMQ \\ \rightarrow \sf \qquad \angle \: OPM = \angle \: OMQ \\ \\ \sf \: but, \qquad \angle \: OPM + \angle \:OMQ = 180 \degree \\ \implies \sf \qquad \angle \: OPM + \angle \: OMQ = 180 \degree \\ \implies \sf \qquad \: 2 \: \angle \: OPM = 180 \degree \\ \implies \sf \qquad \angle \: OPM = 90 \degree \\ \implies \sf \qquad \angle \: OPM = \angle \: OMQ = 90 \degree$

Hence, OM perpendicular to PQ.