Question

O is the centre of the circle and $\angle DAB={50}^o$. Calculate the values of x and y respectively

• A. ${100}^o,{130}^o$
• B. ${80}^o,{100}^o$
• C. ${70}^o,{50}^o$
• D. ${50}^o,{130}^o$

Answer 1

The correct option is A ${100}^o,{130}^o$

$\square ABCD$ is inscribed in a circle.

So, $\square ABCD$ is cyclic quadrilateral.

$\Rightarrow$ $\angle A+\angle C={180}^o$ [ Sum of opposite angles in cyclic quadrilateral are supplementary ]

$\Rightarrow$ ${50}^o+y={180}^o$

$\Rightarrow$ $y={180}^o-{50}^o$

$\therefore$ $y={130}^o$

In $△AOB,$

$\Rightarrow$ $OA=OB$ [ Radius of a circle ]

$\Rightarrow$ $\angle OAB=\angle OBA={50}^o$ [ Base angles of equal sides are also equal ] --- ( 1 )

$\Rightarrow$ $\angle OAB+\angle OBA+\angle AOB={180}^o$

$\Rightarrow$ ${50}^o+{50}^o+\angle AOB={180}^o$ [ From ( 1 ) ]

$\Rightarrow$ $\angle AOB={180}^o-{100}^o$

$\Rightarrow$ $\angle AOB={80}^o$ ------ ( 2 )

$\Rightarrow$ $\angle AOB+\angle BOD={180}^o$ [ Linear pair ]

$\Rightarrow$ ${80}^o+x={180}^o$ [ From ( 2 ) ]

$\Rightarrow$ $x={180}^o-{80}^o$

$\therefore$ $x={100}^o$

$\therefore$ $x={100}^o$ and $y={130}^o.$