Question

A triangle ABC has ∠B = ∠C.
Prove that: The perpendiculars from the mid-point of BC to AB and AC are equal.

Answer 1

Given: A ΔABC in which ∠B = ∠C.
DL is the perpendicular from D to AB
DM is the perpendicular from D to AC

We need to prove that

DL = DM

Proof:

In ΔDLB and ΔDMC

∠DLB = ∠DMC=90⁰  ...[ DL ⊥ AB and DM ⊥ AC ]

∠B=∠C                      ...[ Given ]

BD= DC                    ...[ D is the midpoint of BC ]

∴ By Angel-Angel-SIde Criterion of congruence, 

ΔDLB ≅ ΔDMC

The corresponding parts of the congruent triangles are congruent.

∴DL=DM                ...[ c.p.c.t ]