Question

△ABC and △DBC are two △s on the same base BC. A and D are on opposite side of BC. AD intersects BC at O. If the area(△ABC) is 126 square cm, △DBC is 56 square cm, AO is 24cm then DO =

Answer 1

Answer:

Given:

Two triangles ΔABC and ΔDBC which stand on the same base but on opposite sides of BC.

To Prove:

ar(ΔDBC)

ar(ΔABC)

=

DO

AO

Construction:

We draw AE⊥BC and DF⊥BC.

Proof:

In △AOE and △DOF, we have

∠AEO=∠DFO=90°

∠AOE=∠DOF (Vertically opposite angles)

∴△AOE∼△DOF (By AA criterion of similarity)

⇒

DF

AE

=

DO

AO

…(i)

Now,

ar(ΔDBC)

ar(ΔABC)

=

2

1

×BC×DF

2

1

×BC×AE

⇒

ar(ΔDBC)

ar(ΔABC)

=

DE

AE

…(ii)

From (i) and (ii), we have

ar(ΔDBC)

ar(ΔABC)

=

DO

AO