Answer:
Given parameters
ABCD is a trapezium where AB || DC and diagonals AC and BD intersect at O.
To prove
\(\frac{AO}{BO} = \frac{CO}{DO}\)Construction
Draw a line EF passing through O and also parallel to AB
Now, AB ll CD
By construction EF ll AB
∴ EF ll CD
Consider the ΔADC,
Where EO ll AB
According to basic proportionality theorem
\(\frac{AE}{ED} = \frac{AO}{OC}\) ………………………………(1)Now consider Δ ABD
where EO ll AB
According to basic proportionality theorem
\(\frac{AE}{ED} = \frac{BO}{OD}\) ……………………………..(2)From equation (1) and (2) we have
\(\frac{AO}{OC} = \frac{BO}{OD}\)⇒ \(\frac{AO}{BO} = \frac{OC}{OD}\)
Hence the proof.
