Question

Two sides XY and YZ of the inscribed quadrilateral is equidistant from the center of the circle with radius √32 units. The length of XY is 8 units. Find the angle ∠W.
degrees

Answer 1

Draw a line PY that passes through Y and center of the circle P.
$YP=\sqrt{32}\text{units}$(radius of the circle)
Draw perendiculars from P to the chords XY and YZ.

Given that the chords XY and YZ are equidisant from the centre, hence both chords are equal in length.
$\Rightarrow XY=YZ=8$

Also, the perpendiculars will bisect the chords.
$AY=\frac{XY}{2}\&YB=\frac{YZ}{2}\therefore AY=YB=\frac{8}{2}=4\text{units}$
For $△YAP,$
$A{Y}^2+A{P}^2=Y{P}^2$
$\Rightarrow 4^2+A{P}^2=(\sqrt{32}{)}^2=32$
$\Rightarrow A{P}^2=32-4^2=32-16=16$
$\Rightarrow AP=4\text{units}$

As, $AP=AY,$ the traingle is a isosceles right triangle.
$\Rightarrow \angle AYP={45}^o$

Similarly for $△YBP,$
$\angle BYP={45}^o(\because YB=BP)$
$\angle AYB=\angle AYP+\angle BYP={90}^o$

$\angle W$ is the opposite angle of $\angle AYB$ for the cyclic quadrilateral XYZW.
$\therefore \angle W={180}^o-\angle AYB={90}^o$