1. A=L*l
l=40%L=40L/100=2L/5
A=L*l=L*2L/5=2L²/5=>L²=5A/2=5*160/2=5*80=400=>L=±√400=±20 cm
L>0=>L=20 cm
l=2L/5=2*20/5=2*4=8 cm
P=2L+2l=2*20+2*8=40+16=56 cm
2.M mij.[BC]=>AM->mediana
BM=MC=BC/2=24/2=12
In ΔABC,m(∡BAC)=90°=>(T.Pitagora):AC²=BC²-AB²=24²-12²=576-144=432=>AC=±√432=±12√3 cm
AC>0=>AC=12√3 cm
AM->mediana=>(T.medianei):4AM²=2(AC²+AB²)-BC²=2((12√3)²+12²)-24²=2(432+144)-576=2*576-576=576=>AM²=576/4=144=>AM=±√144=±12 cm
AM>0=>AM=12 cm
fie AO⊥BC=>AO inaltime=>(A doua teoremă a înălțimii):AO*BC=AC*AB<=>24AO=12*12√3<=>2AO=12√3=>AO=6√3 cm
In ΔAOM,m(∡AOM)=90°=>sin(∡AMO)=AO/AM=6√3/12=√3/2=>m(∡AMO)=60°
m(∡AMO)+m(∡AMB)=180°<=>60°+m(∡AMB)=180°=>m(∡AMB)=180°-60°=120°=>m(∡AMB)=120°
