Question

A line `l` is tangent at the end of a diameter `AOB`. Prove that `AOB` bisects any chord parallel to `l`?

Answer 1

Please see below.

Explanation:

Let us consider the figure as given below.

Here we have a circle with centre at `O` and diameter `AOB` with a line `l` as tangent to the circle at `A`. Let us consider a chord `CD`||`l` cutting `AOB` at `Q`. As `CD`||`l`, it is apparent that `CD_|_l`.

Now in `DeltasCOQ` and `OQD`, we have

`OC=OD` `->` both are radius of same circle

`/_OQC=/_OQD=90^@` as `CD`||`l`

`OQ=OQ` `->` being common to two triangles

Hence using RHS, we have `DeltaCOQ-=DeltaOQD`

and hence `CQ=DQ`

i.e. `AB` bisects `CD`

Observe that this can be proved for any chord parallel to `l`.

`Q.E.D.`