ABCD is a trapezium in which AB parallel DC and its diagonals intersect each other at point O . show that AO/BO = CO/DO

Answer:

ABCD is a trapezium in which AB parallel DC and its diagonals intersect each other at point O . show that AO/BO = CO/DO

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.

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