Prove that out of all chords which passesthrough any point circle, tha

1.	Prove that out of all chords which passesthrough any point circle, that chord will besmallest which is perpendicular on diameterwhich passes through that point.
Answer» Let C(O, r) is a circle and let M is a point within it where O is the centre and r is the radius of the circle. Let CD is another chord passes through point M. We have to prove that AB < CD. Now join OM and draw OL perpendicular to CD. In right angle triangle OLM, OM is the hypotenuse. So OM > OL => chord CD is nearer to O in comparison to AB. => CD > AB => AB < CD So all chords of a circle of a circle at a given point within it, the smallest is one which is bisected at that point.

Prove that out of all chords which passesthrough any point circle, that chord will besmallest which is perpendicular on diameterwhich passes through that point.

Answer»

Let C(O, r) is a circle and let M is a point within it where O is the centre and r is the radius of the circle.

Let CD is another chord passes through point M.

We have to prove that AB < CD.

Now join OM and draw OL perpendicular to CD.

In right angle triangle OLM,

OM is the hypotenuse.

So OM > OL

=> chord CD is nearer to O in comparison to AB.

=> CD > AB

=> AB < CD

So all chords of a circle of a circle at a given point within it, the smallest is one which is bisected at that point.