Question

If in a triangle ABC ,sin(a/2)sin(c/2) = sin(b/2) and 2s is the perimeter of the triangle find s

Answer 1

dear student

$$\begin{gathered} \sin \frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{bc}} \\ \sin \frac{B}{2}=\sqrt{\frac{(s-a)(s-c)}{ac}} \\ \sin \frac{C}{2}=\sqrt{\frac{(s-b)(s-a)}{ab}} \\ given \\ \sin \frac{A}{2}\sin \frac{C}{2}=\sin \frac{B}{2} \\ \sqrt{\frac{(s-b)(s-c)}{bc}}\times \sqrt{\frac{(s-b)(s-a)}{ab}}=\sqrt{\frac{(s-a)(s-c)}{ac}} \\ \frac{s-b}{b}=1 \\ s-b=b \\ s=2b \end{gathered}$$

regards