Question-218957

Question Number 218957 by Spillover last updated on 17/Apr/25

Answered by mr W last updated on 17/Apr/25

Commented by mr W last updated on 17/Apr/25

((ar)/2)=14 ⇒a=((2×14)/r) b=((2×15)/r) c=((2×13)/r) s=((a+b+c)/2)=((14+15+13)/r)=((42)/r) Δ_(ABC) =(√(s(s−a)(s−b)(s−c))) 13+14+15=(√((((42)/r))(((14)/r))(((12)/r))(((16)/r))))=((336)/r^2 ) ⇒r=(√((336)/(42)))=2(√2) P=a+b+c=((2×42)/r)=((84)/(2(√2)))=21(√2) ✓

$$\frac{{ar}}{\mathrm{2}}=\mathrm{14}\: \\ $$$$\Rightarrow{a}=\frac{\mathrm{2}×\mathrm{14}}{{r}} \\ $$$${b}=\frac{\mathrm{2}×\mathrm{15}}{{r}} \\ $$$${c}=\frac{\mathrm{2}×\mathrm{13}}{{r}} \\ $$$${s}=\frac{{a}+{b}+{c}}{\mathrm{2}}=\frac{\mathrm{14}+\mathrm{15}+\mathrm{13}}{{r}}=\frac{\mathrm{42}}{{r}} \\ $$$$\Delta_{{ABC}} =\sqrt{{s}\left({s}−{a}\right)\left({s}−{b}\right)\left({s}−{c}\right)} \\ $$$$\mathrm{13}+\mathrm{14}+\mathrm{15}=\sqrt{\left(\frac{\mathrm{42}}{{r}}\right)\left(\frac{\mathrm{14}}{{r}}\right)\left(\frac{\mathrm{12}}{{r}}\right)\left(\frac{\mathrm{16}}{{r}}\right)}=\frac{\mathrm{336}}{{r}^{\mathrm{2}} } \\ $$$$\Rightarrow{r}=\sqrt{\frac{\mathrm{336}}{\mathrm{42}}}=\mathrm{2}\sqrt{\mathrm{2}} \\ $$$${P}={a}+{b}+{c}=\frac{\mathrm{2}×\mathrm{42}}{{r}}=\frac{\mathrm{84}}{\mathrm{2}\sqrt{\mathrm{2}}}=\mathrm{21}\sqrt{\mathrm{2}}\:\:\checkmark \\ $$

Commented by Spillover last updated on 17/Apr/25