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) = 84cm2. Find ar(ΔCOD). 

Answer 1

 

In Δ AOB and COD
∠𝐴𝐵𝑂= ∠𝐶𝐷𝑂 (𝐴𝑙𝑡𝑒𝑟𝑛𝑡𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑖𝑛 𝐴𝐵∥𝐶𝐷)
∠𝐴𝑂𝐵= ∠𝐶𝑂𝐷 (𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑛𝑔𝑙𝑒𝑠) 

By AA similarity criterion, ΔAOB ~ ΔCOD
If two triangles are similar, then the ratio of their areas is equal to the ratio of the squares of their corresponding sides.  

∴ `(area (Δ AOB))/(area(Δ COD))=((AB)/(CD))^2`  

⇒ `84/(area(ΔCOD))=((2CD)/(CD))^2` 

⇒` area (Δ COD)=12 cm^2`