. D, E, F are the midpoints of the sides BC, CA and AB respectively of ABC

Determine the ratio of the areas of DEF and ABC.

Sol: Given  :  D and E are the midpoints of the sides BC and CA respectively of DABC.

To Find     :     Ratio of the areas of DDEF and DABC

Proof  :     Since D and E are the midpoints of the sides BC and CA respectively of DABC

Therefore, DE || BA Þ DE || BF        ..... (i)

Since F and E are the midpoints of AB and AC respectively of DABC.

Therefore, FE || BC Þ FE || BD        ..... (ii)

From equations (i) and (ii), we get that BDEF is a parallelogram.

B = DEF                                        ..... (iii)

[Opposite angles of a parallelogram BDEF]

Similarly AFDE is a parallelogram

ÐA = ÐFDE                                        ..... (iv)

[Opposite angles of a parallelogram BDEF

In DABC and DDEF

B = ÐDEF                                        [From (iii)]

ÐA = ÐFDE                                        [From (iv)]

DABC ~ DDEF                                                                        [By AA similarity]

Since the ratio of the areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides.