Question

Two chords of a circle bisect each other Then they must be

• A. diameters
• B. tangents
• C. secants parallel to each other
• D. none of these

Answer 1

The correct option is A diameters

Two chord AB and CD bisect each other at point O

Join AB,CB,BD and AD

$In△AOCand△BOD$

$OA=OB$

$OC=OD$

$\angle AOC=\angle BOD$ [vertically opposite angle]

$\therefore$By SAS

$△AOC=△BOD$

$\Rightarrow AC=BD$.....1

$In$\triangle AOD$and$\triangle BOC$

$OA=OB$

$OD=OC$

$\angle AOD=\angle BOC$ [vertically opposite angle]

$\therefore$By SAS

$△AOD=△BOC$

$\Rightarrow AD=BC$....2

Adding 1 and 2

$\Rightarrow AC+AD=BD+BC$

$arcDAC=arcCBD$

$\Rightarrow$CD divide the circle in to two semicircle

$\therefore$ CD is a diameter.

Same as AB is a diameter