D, E and F are respectively the midpoints of sides AB, BC and CA of ΔABC. Find the ratio of the areas of ΔDEF and ΔABC.
Answer:
1:4
- In ΔABC, D and F are the midpoints of sides AB and CA respectively.
Therefore, DF||BC [ By midpoint theorem ]
⟹ DF||BE.
Similarly, EF||BD.
Therefore, BEFD is a parallelogram.
⟹∠B=∠EFD,EF=BD=12AB and DF=BE=12BC.
Also, ECFD is a parallelogram.
⟹ ∠EDF=∠C. - Now, in ΔDEF and ΔCAB, we have ∠EFD=∠Band ∠EDF=∠C∴ Thus, the ratio of the areas of \Delta DEF and \Delta ABC is 1:4.