Question

In triunghiul dreptunghic ABC,m (A) =90 grade,se construieste AD perpendicular BC,D apartine( BC )si DE perpendicular pe AB, E aparine lui (AB). Stiind caDE= 6 radical 3 cm si m(<BAD)=30 grade ,calculati PERIMETRUL si ARIA triunghiului ABC.

Answer 1

aplicam teorema ∡ de 30° in triunghiurile ADE, ABD, ADC,ABC
AD=2 x ED⇒AD=12√3
AB=2 x BD
AC=2AD=24√3
BC=2AB=4BD

aplicam pitagora in ABC
AB^2+AC^2=BC^2
4BD^2+24^2 x 3=16BD^2
BD^2=144
BD=12

AB=24 cm
AC=24√3 cm
BC=48 cm

perimetru
P=AB+AC+BC=24+24√3+48=72+24√3=8(9+3√3) cm

aria
A=AB x AC/2=24 x 24√3 / 2=288√3 cm2