Question

Construct a cyclic quardilateral $ABCD$ in which vertical angle $\angle B={65}^o,AB=4cm,AC=6cm$ and $AD=4cm$. Write the steps of construction also.

Answer 1

Given: cyclic quardilateral $ABCD$ in which vertical angle $\angle B={65}^o,AB=4cm,AC=6cm$ and $AD=4cm$
Required: To construct the cyclic quadrilateral.
Construction:
(1) Draw a line segment $AC=6cm$.
(2) Draw a ray $AX$ from point $A$ making an angle equal to $\angle B={60}^o$ downward with line segment $AC$.
(3) Draw another ray $AYbotAX$.
(4) Draw a line $PQ$ perpendicular bisector of line segment $AC$ which intersects the ray $AY$ at a point $O$.
(5) Taking $O$ as centre draw a circle of radius $OA=OC$ which passes through $A$ and $C$.
(6) Taking $A$ as centre draw an arc of radius $AB=4cm$, which intersects the upper part of the circle at $B$.
(7) Taking $A$ as centre draw another arc of radius $AD=4cm$ which intersects the lower part of the circle at $D$.
(8) Join $AB,BC,CD$ and $DA$
Thus the required cyclic quadrilateral is obtained.