Question

AB and XY are parallel chords of a circle. AY intersects BX at O . Prove that OX = OY

Answer 1

the whole proof with diagram is in the image.

Answer 2

Hence Proved.

Step-by-step explanation:

Given,

$AB$ and $XY$ are parallel chords of a circle and $AY$ intersects $BX$ at $O$

Prove that $OX=OY$

∴$\angle OAB=\angle OYX$ and $\angle OBA=\angle OXY$ ( ∵$AB\parallel XY$ )

And,

$\angle AOB=\angle XOY$ (∵ Vertically opposite angle )

Now consider arc $BY$,

Inscribe angles subtended by same arc are equal.

So,

$\angle BAO=\angle OXY$

Similarly, by consider arc $AX$,

$\angle ABO=\angle OYX$

From above,

$\angle OAB=\angle OYX$ and $\angle BAO=\angle OXY$

So,

$\angle OYX=\angle OXY$

∴ $\triangle XOY$ is isosceles triangle.

∴ $OX=OY$

Hence Proved.