MathematicsWhich of the following is the values of x,yand z, where O denotes the centre of the circle?

Which of the following is the values of $x, y$ and $z$ , where $O$ denotes the centre of the circle?

A
$x = 30^{o}, y = 90^{o}$ and $z = 135^{o}$
B
$x = 45^{o}, y = 90^{o}$ and $z = 270^{o}$
C
$x = 30^{o}, y = 60^{o}$ and $z = 270^{o}$
D
$x = 45^{o}, y = 90^{o}$ and $z = 345^{o}$

Solution:

Consider the given figure,

In a cyclic quadrilateral sum of opposite angles are supplementary.
Then from the figure,

3 x + x = 180^{\circ}

\Rightarrow 4 x = 180^{\circ}

\Rightarrow x = 45^{\circ}

The angle subtended by a chord to the circumference of a circle is half of that subtended to the centre.
Then,

\Rightarrow y = 2 \times x

\Rightarrow y = 2 \times 45 °

\Rightarrow y = 90^{\circ}

We know that,

y + z = 360^{\circ}

.

\Rightarrow z = 360^{\circ} - y

\Rightarrow z = 360^{\circ} - 90^{\circ}

\Rightarrow z = 270^{\circ}

Hence, the measure of

x = 45^{\circ}

,

y = 90^{\circ}

and

z = 270^{\circ}

. Therefore, option 2 is correct.