Question:
ABCD is a trapezium such that BC || AD and AD = 4 cm. If the diagonals AC and BD intersect at O such that AOOC=DOOB=12, then BC =
(a) 7 cm
(b) 8 cm
(c) 9 cm
(d) 6 cm
Solution:
Given: ABCD is a trapezium in which BC||AD and AD = 4 cm
The diagonals $\mathrm{AC}$ and $\mathrm{BD}$ intersect at $\mathrm{O}$ such that $\frac{\mathrm{AO}}{\mathrm{OC}}=\frac{\mathrm{DO}}{\mathrm{OB}}=\frac{1}{2}$
To find: DC
In ΔAOD and ΔCOB
∠OAD=∠OCB Alternate angles∠ODA=∠OBC Alternate angles∠AOD=∠BOC Vertically opposite anglesSo, ∆AOD~∆COB AAA similarityNow, correponding sides of similar ∆'s are proportional.AOCO=DOBO=ADBC⇒12=ADBC⇒1
Hence the correct answer is (b)