Question

Draw a circle and any two of its diameters. If you join the ends of these diameters, what is the figure obtained? What figure is obtained if the diameters are perpendicular to each other? How do you check your answer?

Answer 1

Step 1: When diameters are not perpendicular

Let us draw any circle with centre $O$ with any radius

Let $AB$ and $CD$ be the two diameters of this circle.

A quadrilateral is formed when the ends of these diameters are joined as shown below:

We know that,

All diameters of a circle are equal in length, and here the diameters form the diagonals of the quadrilateral.

Hence quadrilateral formed will have its diagonals of equal length.

Also, $OA=OB=OD=OC=$radius of the circle.

Thus, we can say that the diagonals of the quadrilateral bisect each other.

$\Rightarrow$The diagonals of the quadrilateral formed in the circle are equal and bisect each other.

$\Rightarrow$The quadrilateral formed is a rectangle.

Step 2: When diameters are perpendicular

Let us draw any circle with centre $O$ with any radius

Let $DE$ and $GF$ be the two diameters which are perpendicular to each other of this circle.

A quadrilateral is formed when the ends of these diameters are joined as shown below:

We know that, All diameters of a circle are equal in length, and here the diameters form the diagonals of the quadrilateral.

Hence quadrilateral formed will have its diagonals of equal length.

Also, $OA=OB=OD=OC=$radius of the circle.

Thus, we can say that the diagonals of the quadrilateral bisect each other.

And also the diagonals of the quadrilateral are perpendicular to each other.

$\Rightarrow$The diagonals of the quadrilateral formed in the circle are equal and bisect each other at right angles.

$\Rightarrow$The quadrilateral formed is a square.