Question

ABCD is a quadrilateral such that A is the centre of the circle passing through B, C and D. Which one of the following options is correct?

• A. $\angle CBD+\angle CDB=\angle BAD$
• B. $\angle CBD+\angle CDB=\frac{1}{2}\angle BAD$
• C. $\angle CBD+\angle CDB=\frac{2}{3}\angle BAD$
• D. $\angle CBD+\angle CDB=\frac{3}{4}\angle BAD$

Answer 1

The correct option is B $\angle CBD+\angle CDB=\frac{1}{2}\angle BAD$

Given that ABCD is a quadrilateral such that A is the centre of the circle passing through B, C and D. Join AC.



Since angle subtended by an arc at the centre is double the angle subtended by it at point on the remaining part of the circle,

$\angle CAD=2\times \left(\angle CBD\right)$$\dots$(1) and
$\angle BAC=2\times \left(\angle CDB\right)$$\dots$(2).

Adding (1) and (2), we get
$\angle CAD+\angle BAC=2(\angle CBD+\angle CDB)$
$\Rightarrow \angle BAD=2(\angle CBD+\angle CDB)$
Hence, $\angle CBD+\angle CDB=\frac{1}{2}\angle BAD.$