Question : Through the midpoint M of side CD of a parallelogram ABCD, the line BM is drawn intersecting AC in L and AD produced in E. Prove that EL=2BL
Doubt by Mayank
Solution:
Given : ABCD is a parallelogram.
M is the mid point of CD i.e. MC=MD
To Prove : EL=2BL
Proof :
In ∆BMC and ∆EMD
∠1=∠2 (Alternate Interior Angles)
∠3=∠4 (Vertically opposite angles)
MC=MD (Given)
∆BMC≅∆EMD (By AAS)
BC=ED (By CPCT) —(1)
Also,
BC=AD (Opposite sides of ||gram are equal) — (2)
Now,
In ∆AEL and ∆CBL
∠1=∠2 (Alternate Interior Angles)
∠5=∠6 (Vertically opposite angles)
∆AEL~∆CBL (By AA Similarity Crtiteria)
EL/BL=AE/CB (By CPST)
EL/BL=(AD+ED)/CB
EL/BL= (BC+BC)/BC [From (1) and (2)]
EL/BL=2BC/BC
EL/BL=2
EL=2BL
Hence proved.