Question

If in a $△ABC$, the altitudes from the vertices $A$, $B$ and $C$ on opposite sides are in HP, then $\sin A$, $\sin B$, $\sin C$ are in

• A. HP
• B. Arithmetic-geometric progression
• C. AP
• D. GP

Answer 1

The correct option is C AP

In $△BDA$,
$\cos ({90}^o-B)=\frac{AD}{c}$
$\Rightarrow AD=c\sin B$
Similarly, $BE=a\sin C$ and
$CF=b\sin A$
Since, $AD,BE,CF$ are in HP.
$\therefore c\sin B,a\sin C,b\sin A$ are in HP

$\Rightarrow \frac{1}{\sin C\sin B},\frac{1}{\sin A\sin C},\frac{1}{\sin B\sin A}$ are in AP

$\Rightarrow \sin A,\sin B,\sin C$ are in AP .....(multiply by $\sin A\sin B\sin C$)