Question

If AC and BD are the two chords of a circle which bisect each other, then the quadrilateral ABCDis a

• A. rectangle
• B. square
• C. parallelogram
• D. none of these

Answer 1

Answer: (A)

rectangle

$\Rightarrow$ In given figure, AC and BD are the chords of a circle which bisect each other at the point O.

Now in $△$OAB and $△$ OCD

$\Rightarrow$ OA = OC (Given)

$\Rightarrow$ OB = OD (Given)

$\Rightarrow$ $\angle$AOB = $\angle$COD (Vertically opposite angles)

So by SAS congruent criteria,

$\Rightarrow$ $△$OAB $\cong$ $△$OCD

$\Rightarrow$ AB = CD (By CPCT) ---- (1)

Similarly we can show that,

$\Rightarrow$ 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${}^{\circ }$

So, $\angle$A = 90${}^{\circ }$ and $\angle$C = 90${}^{\circ }$ ----- (3)

Similarly we can show that,

$\angle$B = 90${}^{\circ }$ and $\angle$D = 90${}^{\circ }$ -------- (4)

From equation (3) and (4), we get

$\Rightarrow$ $\angle A=\angle B=\angle C=\angle D={90}^{\circ }$

So ABCD is a rectangle.