Question

If a line segment joining mid-points of two chords of a circle passes through the center of the circle, prove that the two chords are parallel?

Answer 1

Proof the statement.

Let $AB$ and $CD$ be the given chords.

Let $L$ is the midpoint of chord $AB$ and $M$ is the midpoint of chord $CD$.

Let's draw the figure.

Let $PQ$ is the line passing through these midpoints and the center of the circle.

We can see that from the figure, $PQ$ is the diameter of the circle.

From the figure, as $L$ and $M$ are midpoints of $AB$ and $CD$ respectively.

$AB\perp OL$ and $CD\perp OM$,

$$\begin{gathered} \therefore \angle OLA=90° \\ \angle OMD=90° \end{gathered}$$

As they are equal and alternate angles, $AB\parallel CD$.

Hence, proved that the two chords are parallel.