Question

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

$B=\frac{\pi }{8}$ and $C=\frac{5\pi }{8}.$ The altitude from $A$ to the side $BC$ is?

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

Answer 1

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

Applying sine rule we get
$\frac{a}{\sin \frac{\pi }{4}}=\frac{b}{\sin \frac{\pi }{8}}=\frac{c}{\sin \frac{5\pi }{8}}$

Hence

$c=\sin \frac{5\pi }{8}.\sqrt{2}a$

$=\sin \frac{3\pi }{8}.\sqrt{2}a$

$=a\sqrt{2}.\cos \left(\frac{\pi }{8}\right)$...(i)

Now from the given figure,

$AD=c\sin B$

$=c\sin \left(\frac{\pi }{8}\right)$

$=a\sqrt{2}.\cos \left(\frac{\pi }{8}\right).\sin \frac{\pi }{8}$

$=\frac{a\sqrt{2}}{2}.\sin \left(\frac{\pi }{4}\right)$

$=\frac{a}{2}$