In trapezium ABCD, BC \(\parallel\) AD and AD = 4 cm. The diagonals AC and BD intersect at point O in such a way that \(\frac{AO}{OC} = \frac{DO}{OB} = \frac{1}{2}\). Find the length of BC.

Q.In trapezium ABCD, BC \(\parallel\) AD and AD = 4 cm. The diagonals AC and BD intersect at point O in such a way that \(\frac{AO}{OC} = \frac{DO}{OB} = \frac{1}{2}\). Find the length of BC.

In triangles AOD and BOC: \(\angle\)AOD = \(\angle\)BOC [Vertically opposite angles] And \(\frac{AO}{OC} = \frac{DO}{OB}\) \(\therefore\) \(\triangle\)AOD ~ \(\triangle\)BOC \(\therefore \frac{AO}{OC} = \frac{DO}{OB} = \frac{AD}{BC} = \frac{1}{2}\) i.e., \(\frac{AD}{BC} = \frac{1}{2}\) i.e., \(\frac{4}{BC} = \frac{1}{2}\) i.e., BC = 8 \(\therefore\) The length of BC is 8 cm.