## State and prove Basic Proportionality Theorem. Using the above theorem, if ABCD is a trapezium whose diagonals intersect each other at O sho

Question

State and prove Basic Proportionality Theorem. Using the above theorem, if ABCD is a trapezium whose diagonals intersect each other at O show that AO/OC = BO/OD.​

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4 days 2021-10-14T00:16:50+00:00 2 Answers 0 views 0

## Answers ( )

1. 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

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

$$\frac{AO}{OC}$$ = $$\frac{BO}{OD}$$

Hence the proof.

2. Given parameters

ABCD is a trapezium where AB || DC and diagonals AC and BD intersect at O.

To prove

AO/BO=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

Where EO ll AB

According to basic proportionality theorem

Given parameters

AE/ED=AO/OC………………………………(1)

Now consider Δ ABD

where EO ll AB

According to basic proportionality theorem

AE/ED=BO/OD……………………………..(2)

From equation (1) and (2) we have

AO/OC=BO/OD

Hence the proof.