In the figure ABCD is a parallelogram and E divides BC in the ratio 1:3. DB and AE intersect at F. Show that DF=4FB and AF=4FE.

Question

In the figure ABCD is a parallelogram and E divides BC in
the ratio 1:3 DB and AE intersect at F Show that DF=4FB and
AF=4FE

Ashwini Kumar · Accepted Answer

We have,E divides BC in 1:3∴BEEC=13⇒BEBC=14Now, ABCD is a parallelogram∴BC=ADHence, BEAD=14     
1Now,BC and AD are parallel lines and AE is a transversal line,∴∠DAF=∠FEBSimilalry, BC and AD are parallel lines and BD is a transversal line,∴∠ADF=∠EBF∴△AFD~△EFB       By A-A similarity∴BFDF=EFAF=BEAD⇒BFDF=EFAF=14⇒BFDF=14⇒DF=4BFAnd⇒EFAF=14⇒AF=4EFHence, DF=4FB and AF=4FE