Question

ABCD is a trapezium such that BC || AD and AD = 4 cm. If the diagonals AC and BD intersect at O such that \[\frac{AO}{OC} = \frac{DO}{OB} = \frac{1}{2}\], then BC =

Options

• 7 cm

• 8 cm

• 9 cm

• 6 cm

Answer 1

Given: ABCD is a trapezium in which BC||AD and AD = 4 cm

The diagonals AC and BD intersect at O such that `(AO)/(OC)=(DO)/(OB)=1/2`

To find: DC

 In ΔAOD and ΔCOB

\[\angle OAD = \angle OCB \left( \text{Alternate angles} \right)\]
\[\angle ODA = \angle OBC \left( \text{Alternate angles} \right)\]
\[\angle AOD = \angle BOC \left( \text{Vertically opposite angles} \right)\] 
\[So, ∆ AOD~ ∆ COB \left( \text{AAA similarity} \right)\]
\[\text{Now, correponding sides of similar ∆ 's are proportional} . \]
\[\frac{AO}{CO} = \frac{DO}{BO} = \frac{AD}{BC}\]
\[ \Rightarrow \frac{1}{2} = \frac{AD}{BC}\]
\[ \Rightarrow \frac{1}{2} = \frac{4}{BC}\]
\[ \Rightarrow BC = 8 cm\]
Hence the correct answer is `b`