Q. 245.0( 3 Votes )

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

Answer :

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


To prove: BE: EX = 3: 1


Theorem Used:


1.) If two corresponding angles of two triangles are equal, the triangles are said to be similar.


2.) Basic proportionality theorem:


If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides into same ratio.


Proof:


Const: Through D, Draw DF||BX



In ΔEAX and Δ ADF


EAX = ADF (Common)


AXE = DAF (Corresponding angles)


Then, ΔEAX ~ Δ ADF


So,


(Corresponding parts of similar triangle are proportion)


(AE = ED given)


DF = 2EX. ……………(i)


In ΔCDF and ΔCBX (By AA similarity)


SO, (Corresponding parts of similar triangle area proportion)


(BD = DC given)


BE + EX = 2DF


BE = EX = 4EX


BE = 4EX – EX


BE = 4EX – EX


BE = 3EX



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