Question

In a trapezium ABCD, it is given that AB || CD and AB = 2CD. Its diagonals AC and BD intersect at the point O such that ar(ΔAOB) = 84 cm². Find ar(ΔCOD).

Answer 1

In triangle COD and AOB,

∠OCD = ∠OAB 

(Alternate angles as AB || CD) 

∠CDO = ∠OBA 

(Alternate angles as AB || CD) 

∠COD = ∠AOB 

(Vertically opposite angles) 

∴ ∠COD = ∠AOB 

(By AAA similarity criteria)

∴ `(ar(ΔCOD))/(ar(ΔAOB)) = ((CD)/(AB))^2`

(The ratio of the areas of two similar triangles Is equal to square of the ratio of thelr corresponding sides)

⇒ `ar(ΔCOD) = ((CD)/(AB))^2 ar(ΔAOB)`

(∵ AB = 2CD (Given))

= `1/4 xx 84`

= 21 cm²

(∵ ar(ΔAOB))

= 84 cm² (Given))

∴ Area of ΔCOD is 21 cm².