Question

ABC is a triangle D, E are the mid midpoints of AB and BC. Then find the ratios of the areas of triangle ABC and ADE

Answer 1

Let F be the middle point of AC.

the triangles BDE, ADF, EFC, DEF are all similar to the triangle ABC.  But their sides are half of those of triangle ABC.

Thus all the above triangles are all equivalent/congruent.  

Let us consider the parallelogram  ADEF.  Its area is twice that of triangle ADF or DEF.   But the area of the triangle ADE is half of the parallelogram ADEF, as AE is its diagonal.  Thus area of triangle ADE = area of triangle DEF.

Hence, the ratio is  1/4.    

It will be same for triangle  CDF or CDE, BFD or BFE.