In ΔEAD dreptunghic:
sin BAD = ED/AD = sin 30° = 1/2
ED/AD = 1/2
6√3 / AD = 1/2
AD = 12√3 cm
In ΔBAD dreptunghic in D:
tg BAD = BD/AD
tg 30° = BD/12√3
√3/3 = BD /12√3
BD = 12√3 x √3/3 = 12 cm
In ΔADC dreptunghic in D:
∡DAC = ∡BAC - ∡BAD = 90 - 30 = 60°
tg DAC = DC/AD
tg 60 = DC/12√3
√3 = DC/12√3
DC = 12√3 x √3
DC = 12 x 3 = 36 cm
BC = BD + DC = 12 + 36 = 48 cm
∡ACB = 90 - ∡DAC = 90 - 60 = 30°
In ΔABC
sin ACB = AB/BC
sin 30 = AB/48
1/2 = AB/48
AB = 24 cm
cos ACB = AC/BC
cos 30 = AC/48
√3/2 = AC/48
AC = 24√3 cm
Acum putem calcula aria si perimetru ΔABC.
P = AB + AC + BC = 24 + 24√3 + 48 = 72 + 24√3 = 24√3(√3 + 1) cm
A = AD x BC / 2 = 12√3 x 48/2 = 12√3 x 24 = 288√3 cm²
