D is the mid-point of side BC of a ∆ABC. AD is bisected at the point E and BE produced cuts AC at the point X. Prove that BE = EX = 3 : 1
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SOLUTION :
GIVEN : In ∆ABC , D is the midpoint of BC, E is the midpoint of AD. BE produced meets AC at X.
To prove : BE: EX = 3 : 1
PROOF :
In Δ BCX and ΔDCY
∠CBX = ΔCBY (corresponding angles)
∠CXB = ΔCYD (corresponding angles)
ΔBCX∼ΔDCY (AA similarity)
BC/DC = BX/ DY = CX/CY
[Since, corresponding sides of two similar triangles are proportional]
BX/DY = BC/DC
BX/DY = 2DC/DC
[As D is the midpoint of BC]
BX/DY = 2/1………...(i)
In ΔAEX and ΔADY,
∠AEX = ΔADY (corresponding angles)
∠AXE = ΔAYD (corresponding angles)
ΔAEX ∼ ΔADY (AA similarity)
AE/AD = EX/DY = AX/AY
[Since, corresponding sides of two similar triangles are proportional]
EX/DY = AE/AD
EX/DY = AE/2AE (As D is the midpoint of BC)
EX/DY = ½…………..…(ii)
On Dividing eqn. (i) by eqn. (ii)
BX/EX = 4/1
BX = 4EX
BE + EX = 4EX
BE = 4EX - EX
BE = 3EX
BE /EX = 3/1
BE : EX = 3:1
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