Question

If D, E, F are the respectively the midpoints of sides BC, CA and AB of ΔABC. Find the ratio of the areas of ΔDEF and ΔABC. 

Answer 1

 

By using mid theorem i.e., the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.
∴ DF || BC 

ANd `DF=1/2 BC` 

⟹ DF = BE  

Since, the opposite sides of the quadrilateral are parallel and equal.
Hence, BDFE is a parallelogram
Similarly, DFCE is a parallelogram.
Now, in ΔABC and ΔEFD 

∠𝐴𝐵𝐶= ∠𝐸𝐹𝐷 (𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑜𝑓 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚)
∠𝐵𝐶𝐴=∠𝐸𝐷𝐹 (𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑜𝑓 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚)  

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

∴ `(area(Δ DEF)) /(area(ΔABC))=((DF)/(BC))^2=((DF)/(2DF))^2=1/4` 

Hence, the ratio of the areas of ΔDEF and ΔABC is 1 : 4.