Question

Draw a circle and any two of its (non- perpendicular) diameter. 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 the answer?

Answer 1

We know that angle in a semicircle is ${90}^o$

Consider the case of two non perpendicular diameters

$\therefore \angle ABC=\angle ADC={90}^o$

Similarly, $\angle DAB=\angle DCB={90}^o$

So, in $ABCD$ all angles of vertices are ${90}^o$

Thus, $ABCD$ is a rectangle.

Now consider the case of perpendicular diameters

If $\angle AOB=\angle BOC={90}^o$

$AO=CO,OB=OC$..... (radius of circle)

then $\Delta AOB\cong \Delta COB$

Thus $AB=CB$ (by cpet),

Hence $ABCD$ becomes square.

Thus,

$\left(i\right)$ On joining the ends of any two diameters of the circle, the figure obtained is a rectangle.
$\left(ii\right)$ On joining the ends of any two diameters of the circle, perpendicular to each other, the figure obtained is a square.