Question

Construct a cyclic quadrilateral XYZW in which WX=3.5cm, $\mathrm{\angle }X=75{}^{\circ }$, XY=2.8cm, and the vertex Z lies on the angle bisectors of $\mathrm{\angle }WXY$.

Answer 1

Hint: In this question we are asked to construct a cyclic quadrilateral. For that we should be aware about the basic geometrical construction steps. We will create the quadrilateral step by step with the help of a ruler and a compass. The use of a protractor should also be known because we would need to make cuts at several spots at certain angles.

Complete step by step answer:
A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle. It is also sometimes called inscribed quadrilateral. The circle which consists of all the vertices of any polygon on its circumference is known as the circum-circle or circumscribed circle.
First, draw a line segment WX which has length 3.5cm. After that pointing from X, make an angle of  $75{}^{\circ }$ and identify a point Y at a distance 2.8cm from X i.e. XY=2.8cm.

 Now, draw the perpendicular bisector of XY and WX. The bisectors of XY and WX will meet at a new point O.

Now, taking O as the centre, draw a circle of radius OX. Now, construct the angular bisector of $\mathrm{\angle }WXY$. This angular bisector will cut the circle constructed earlier at Z.

Lastly, join the points W,X,Y and Z. This quadrilateral constructed would be the required quadrilateral.
The constructed diagram looks like follows:

Note: While constructing the quadrilateral, make sure that you take the accurate distance whenever you are making a mark or identifying a new point on the plane. Also, you should be aware about the use of a protractor, make sure that you mark the angle at the correct spot otherwise that would lead to an invalid figure.