Practice problems

Q1. If the medians BE and CF are equal in a △ABC prove that AB=AC. 
S1.

Let G be the centroid and let CG = 2x = BG.

Let EG = FG = x.

Now triangles EGC and FGB are congruent by SAS. Why?

Since EG = FG, CG = BG and angle FGB = angle CGE because they are vertical angles.

=> FB = EC

And BF = FA since F is midpoint of AB, similarly AE = EC.

=> AF + FB = AE + EC => AB = AC H.P. Q2. Let (D, E, F) be the feet of the altitude from (A, B, C) in a (\triangle ABC). Prove that the perpendicular bisector of (EF) also bisects (BC).

S2. BCEF is a cyclic quadrilateral with BC as diameter. Perpendicular bisector of EF will pass through the center of the circle since EF is a chord. Center lies on the midpoint of BC as it is a diameter. Hence perpendicular bisector of EF will pass through the midpoint of BC. H.P.

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