Question

$AC$ and $BD$ are chords of a circle which bisect each other. Prove that $ABCD$ is a rectangle.

Answer 1

Hint: In this question there is a property of cyclic quadrilateral i.e. the sum of the opposite angles of a cyclic quadrilateral is ${180^ \circ }$ will be used to prove the statement that $ABCD$ is a rectangle.

Complete step-by-step answer:

According to the question, $AC$ and $BD$ are chords of a circle which bisect each other and we have to prove that $ABCD$ is a rectangle.
Let $AC$ and $BD$ are chords of a circle which bisect each other at the point O.
Now in $\Delta OAB$ and $\Delta OCD$
$OA = OC$(given)
$OB = OD$(given)
$\angle AOB = \angle COD$ (since vertically opposite angles)
So by SAS congruent criteria,
$\Delta OAB \cong \Delta OCD$
By $CPCT$
$AB = CD...........1$
Similarly we can show that
$AD = CB............2$
Add equation $1$ and $2$, we get
       $AB + AD = CD + CB$
$ \Rightarrow \angle BAD = \angle BCD$
So $BD$ divides the circle into two equal semi-circle and the angle of each one is $90$
So $\angle A = 90$and $\angle C = 90.............3$
Similarly we can show that
$\angle B = 90$and $\angle D = 90................4$
From equation $3$and $4$, we get
$\angle A = \angle B = \angle C = \angle D = 90$
So $ABCD$ is a rectangle.

Note: In such types of questions first draw the figure of cyclic polygon which has circumscribed circle and according to figure see what basic properties of cyclic quadrilateral can be used . Hence it is advisable to remember some basic properties while involving cyclic polygon questions .