Question

Calculati raza cercului inscris in triunghiul ABC daca:
a) AB=20 cm , BC=12 cm si AC= 16 cm ;
b)AB= 12 cm, BC= 13 cm si AC= 15 cm.

Answer 1

$Vom~utiliza~formula~\boxed{S=p*r},~unde~S~este~aria~triunghiului, \\ p-semiperimetrul~triunghiului~si~r-raza~cercului~inscris~ \\ triunghiului.$

$a)~p= \frac{20+12+16}{2}=24~(cm). \\ \\ S= \sqrt{p(p-a)(p-b)(p-c)} = \sqrt{24(24-12)(24-16)(24-20)}= \\ = \sqrt{24*12*8*4}= \sqrt{24*24*4*4}=24*4=96~(cm^{2}). \\ \\ S=p*r \Rightarrow r= \frac{S}{p}= \frac{96}{24}=4~(cm).$

$b)~p= \frac{12+13+15}{2}= 20~(cm). \\ \\ S= \sqrt{p(p-a)(p-b)(p-c)}= \sqrt{20(20-13)(20-15)(20-12)} = \\ = \sqrt{20*7*5*8} = 20 \sqrt{14}~ (cm^2). \\ \\ S=p*r \Rightarrow r= \frac{S}{p}= \frac{20 \sqrt{14} }{20}= \sqrt{14}~(cm).$

$***~In~ambele~cazuri~am~notat~a=BC;~b=AC~si~c=AB~***$

Edit: "r- lungimea razei cercului inscris triunghiului".