Question

In a ABC,Angle A=90°,AB=AC,Bisector of angle A meet BC at D. Prove that BC=2AD.Using SAS criterion

Answer 1

in triangle ACD and ABD


AC =AB (given)

angle CAD = angle DAB
( both 45° as AD is bisector of angle A)

AD = AD ( common)

so, by SAS congruence creteria triangle ACD is congruent to triangle ABD.

then,
CD = DB (by c.p.c.t.)


angle A + angle B + angle C = 180°
90° + angle B + angle C = 180°
angle B + angle C = 90°

as we know that,
CA = BA
then the angles at base would also be equal so,
angle C = angle B = 45°


CB = CD + DB ---------(i)

as we know that
angle BAD = angle ADB (both 45°)
then,
AD = DB ( equal sides opposite to equal angles)

again,
angle ACD = angle DAC (both 45°)
then,
AD = CD (equal sides opposite to equal angles)

substitute for the in (i)

CB = AD + AD
CB = 2AD.