Question

1 in in triunghiul dreptunghic ABC , m < A = 90 , se stie ca BC = 12 rad 13 cm , iar AB \ AC = 2\3 calculati a ) lungimile catetelor triunghilui dat b ) aria triunghilui
2 in triunghil dreptunghic ABC , m < A = 90 grade , se stie ca AB \ AC = #\4 , iar inaltimea AD =24 cm , D apartine (BC) . cLCULATI :
A ) perimetrul trunghilui
B ) aria trunghilui

Answer 1

1
AB/AC=2/3 = 2k/3k
BC²=AB²+BC²
(12√13)²=(2k)²+(3k)²
144*13=4k²+9k²
144=k²
k=12
AB=2*12=24
AC=3*12=36
Atriunghi= 24*36/2=432 cm²

2.
AB/AC=3/4=3k/4k
BC=√(3k)²+(4k)²=√25k²=5k
notam BD=x  si DC=5k-x
AD²=(3k)²-x² =24² 
AD²=(4k)²-(5k-x)² 
(3k)²-x² =(4k)²-(5k-x)²
9k²-x²=16k²-25k²+10kx-x²
18k²=10kx
9k=5x=  ⇒ k=5x/9
9*(5x/9)²-x² =576 
25x²/9-x²=576
25x²-9x²=5184
16x²=5184
x²=324
x=18
k=5x/9=5*18/9=10
AB=3*10=30
AC=4*10=40
BC=5k=5*10=50
Perimetrul=P=30+40+50=120 cm
Atriunghi= 24x50/2=600 cm²