Question

The diagonals of a trapezium ABCD, in which AB parallel DC intersect at O. If AB=2CD, then find the ratio of areas of triangles AOB and COD.

Answer 1

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Answer 2

Given :-

• Diagonals of a trapezium ABCD with AB || CD, intersect each other at point 'O' .
• AB = 2CD .

To Find :-

• Area ∆AOB : Area ∆COD .

Solution :-

from image we have,

→ ABCD is a trapezium .

→ AB ∣∣ DC .

now, in ∆AOB and ∆COD , we have,

→ ∠ABO = ∠CDO (Alternate angles.)

→ ∠BAO = ∠DCO (Alternate angles.)

→ ∠AOB = ∠COD (Vertically opposite angles.)

so,

→ ∆AOB ~ ∆COD (By AAA Similarity.)

now, we know that,

• If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.

then,

→ Ar.∆AOB : Ar.∆COD = AB² : CD²

→ ∆AOB / ∆COD = (AB/CD)²

given that, AB = 2CD

→ Ar.∆AOB / Ar.∆COD = (2CD/CD)²

→ Ar.∆AOB / Ar.∆COD = (2/1)²

→ Ar.∆AOB / Ar.∆COD = 4 / 1

→ Ar.∆AOB : Ar.∆COD = 4 : 1 (Ans.)

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