Question

In triunghiul ABC se cunosc:AC=2,5 cm,m(<A)=75 grade si m(<C)=60 grade.

Aflati

a)lungimea inaltimii triunghiului ABC,corespunzatoare laturii (BC)
b)perimetrul triunghiului ABC

Answer 1

a) Fie D∈[BC] a.i. AD _|_ BC.

m(<ACD)=60* si m(<ADC)=90* => m(<CAD)=30*.

sin(<ACD)=$\frac{AD}{AC}$ => AD= AC* sin(<ACD)= 2,5*sin(60*)=$\frac{5}{2}* \frac{ \sqrt{3} }{2}= \frac{5 \sqrt{3} }{4}(cm).$

b) m(<BAD)=m(<BAC)-m(<CAD)=75*-30*=45*.

Triunghiul BAD este dreptunghic (in D) isoscel => AD=BD si $AB= AD\sqrt{2}$.

AD=BD => BD=$\frac{ 5\sqrt{3} }{4}$ cm.
$AB= AD\sqrt{2}= \frac{ 5\sqrt{6} }{2}(cm).$

In ΔACD-dreptunghic in D, [CD] se opune unghiului de 30* => $CD= \frac{AC}{2} = \frac{5}{4} (cm).$

BC=CD+BD=$\frac{5}{4}+ \frac{5 \sqrt{3} }{2} = \frac{5+ 10\sqrt{3} }{4} (cm).$

$P_{ABC}= AB+BC+AC= \frac{ 5\sqrt{6} }{2}+ \frac{5+10 \sqrt{3} }{4}+ \frac{5}{2}= \frac{10 \sqrt{6}+15+10 \sqrt{3} }{4} (cm).$