In the adjoining figure, △ABC is an isosceles triangle in which AB = AC. If E and F be the midpoints of AC and AB respectively, prove that BE = CF. Hint. Show that ,△BCF≅△CBE.
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Given Δ is an isosceles triangle
⇒ AB-BC _______(1)
and ∠B=∠C ________(2)
Here E andF are midpoints of AC and AB respectively
∴ AF = FB and AE = EC
know , AB = BC
⇒ AF+FB = AE + EC
⇒ 2AF = 2AE
⇒ AF = AE.
⇒ AF =FB = AE =EC _______(3)
In ΔBCF and Δ CBE
BC = BC [common side]
∠B=∠C [from (2)]
BF = EC [from (3)]
By SAS condition for congruency.
ΔBCF≅ΔCBE.
∴ since ΔBCF≅Δ CBE, by properly of congruncy we can with that
BE = CF.
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