Question

Question 194
Draw a circle of radius 3 cm and draw its diameter and label it as AC. Construct its perpendicular bisector and let it intersect the circle at B and D. What type of quadrilateral is ABCD? Justify your answer.

Answer 1

Steps of Construction
Step I Taking O as centre and radius=OC = 3 cm, draw a circle.
Step II Join A to C and draw a perpendicular bisector of AC that cuts the circumference of the circle at B and D.
Step III Join B and D.
Step IV Thus, ABCD is a cyclic quadrilateral.

Justification:
In quadrilateral ABCD,
$\angle B=\angle D={90}^{\circ }$ [angle in a semi-circle]
$\angle A=\angle C={90}^{\circ }$
$\Rightarrow$$\angle B+\angle D={180}^{\circ }$
and $\angle A+\angle C={180}^{\circ }$
$\Rightarrow$opposite angles are supplementary
Since, opposite angles are supplementary, thus the quadrilateral ABCD is a cyclic quadrilateral.