D is the mid-point of side BC of a . AD is bisected at the point E and BE produced cuts AC at the point X. Prove that BE : EX = 3 : 1.

Question

Philoid · Accepted Answer

Given:- In ΔABC, D is the midpoint of BC and E is the midpoint of AD.

To prove:- BE: EX = 3 : 1

Const:- Through D, Draw DF||BX

Prof:- In ΔEAX and Δ ADF

Then, ΔEAX ~ Δ ADF

So, (Corresponding parts of similar triangle are proportion)

Or, (AE = ED given)

Or, DF = 2EX. ……………(i)

In ΔCDF and ΔCBX (By AA similarity)

SO, (Corresponding parts of similar triangle area proportion)

Or (BD = DC given)

Or BE + EX = 2DF

Or BE = EX = 4EX

Or BE = 4EX – EX

Or BE = 3EX

Or BE/EX =3/1