Question

Prove that the line joining the mid-point of a chord to the centre of the circle passes through the mid-point of the corresponding minor arc.

Answer 1

Given: A circle with centre $O$ and a chord $AB$

Let $M$ be the midpoint of $AB$ and $OM$ is joined and produced to meet the minor are $AB$ at $N$

To prove: $N$ is the midpoint of arc $AB$

Construction: Join $OA,OB$

Proof :Since $M$ is mid point of $AB$.

$\therefore OM\perp AB$

In $\Delta OAM\text{and}\Delta OBM$

$OA=OB$ (Radii of the circle)

$OM=OM$ (common)

$AM=BM$ ($M$ is mid point of $AB$)

$\therefore \Delta OAM\cong \Delta OBM$ (SSS criterian)

$\therefore \angle AOM=\angle BOM$ (Corresponding parts of congruent triangles)

$\Rightarrow \angle AON=\angle BON$

But these are centre angles at the centre made by arcs $AN$ and $BN$

Hence $N$ divides the are in two equal parts