In $ riangle A B C$, if $a=13, b=14$ and $\cos \frac{C}{2}=\frac{3}{\sqrt{13}}$, then $2 r_1=$

Question

In $	riangle A B C$, if $a=13, b=14$ and $\cos \frac{C}{2}=\frac{3}{\sqrt{13}}$, then $2 r_1=$

Aicharya · Accepted Answer

$\begin{aligned} & 	ext { } a=13, b=14, \cos \frac{C}{2}=\frac{3}{\sqrt{13}} \ & s=\frac{a+b+c}{2} \Rightarrow s=\frac{27+c}{2}\end{aligned}$ $\begin{aligned} & \cos \frac{C}{2}=\frac{3}{\sqrt{3}} \Rightarrow \sqrt{\frac{s(s-c)}{a b}}=\frac{3}{\sqrt{3}} \ & \Rightarrow s(s-c)=126 \Rightarrow\left(\frac{27+c}{2}ight)\left(\frac{27-c}{2}ight)=126 \ & \Rightarrow 729-c^2=504 \Rightarrow c=15 	ext { and } s=21\end{aligned}$ $\begin{aligned} & \cos \frac{A}{2}=\sqrt{\frac{s(s-a)}{b c}}=\sqrt{\frac{21 	imes 8}{14 	imes 15}} \ & \cos \frac{B}{2}=\sqrt{\frac{s(s-b)}{a c}}=\sqrt{\frac{21 	imes 7}{13 	imes 15}} \ & 2 r_1=\frac{2 a \cos \frac{B}{2} \cos \frac{C}{2}}{\cos \frac{A}{2}}=\frac{2 	imes 13 	imes \sqrt{\frac{21 	imes 7}{13 	imes 15}} 	imes \frac{3}{\sqrt{13}}}{\sqrt{\frac{21 	imes 8}{14 	imes 15}}}=21=s\end{aligned}$

In $\triangle A B C$, if $a=13, b=14$ and $\cos \frac{C}{2}=\frac{3}{\sqrt{13}}$, then $2 r_1=$

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