Question

In the given figure, MN || BC and AM : MB = 1 : 2. 

Find `("area"(ΔAMN))/("area"(ΔABC))`.

Answer 1

We have
AM : MB = 1 : 2 

⇒ AM:MB=1:2 

⇒ `(MB)/(AM)=2/1` 

Adding 1 to both sides, we get 

⇒`( MB)/(AM)+1=2/1+1` 

⇒`(MB+AM)/(AM)=(2+1)/1` 

⇒ `(AB)/(AM)=3/1` 

Now, In ΔAMN and ΔABC
∠𝐴𝑀𝑁 = ∠𝐴𝐵𝐶 (𝐶𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒𝑠 𝑖𝑛 𝑀𝑁 ∥ 𝐵𝐶)
∠𝐴𝑁𝑀 = ∠𝐴𝐶𝐵 (𝐶𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒𝑠 𝑖𝑛 𝑀𝑁 ∥ 𝐵𝐶)
By AA similarity criterion, ΔAMN ~ Δ ABC
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 AMN))/(area(ΔABC))=((AM)/(AB))^2=(1/3)^2=1/9`