Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively.

If $B=\frac{\pi }{8}$ and $C=\frac{5\pi }{8}$, then length of altitude $\left(p\right)$ dropped from $A$ is?

• A. $\frac{a}{\sqrt{2}}$
• B. $2a$
• C. $\frac{1}{2}(b+c)$
• D. None of these

Answer 1

The correct option is A $\frac{a}{\sqrt{2}}$

$\frac{1}{2}pa=$ area $=\frac{1}{2}bc.\sin \frac{\pi }{4}$

$\therefore p=\frac{bc}{a\sqrt{2}}$

Also, $\frac{a}{\sin \frac{\pi }{4}}=\frac{b}{\sin \frac{\pi }{8}}=\frac{c}{\sin \frac{5\pi }{8}}$

$\Rightarrow b=\sqrt{2}a\sin \frac{\pi }{8},c=\sqrt{2}a\sin \frac{5\pi }{8}$

$\therefore p=\frac{2a^2\sin \frac{\pi }{8}\sin \frac{5\pi }{8}}{a\sqrt{2}}=\frac{a}{\sqrt{2}}[\because (\cos \frac{4\pi }{8}-\cos \frac{6\pi }{8})=\frac{1}{\sqrt{2}}]$