Question

If in $△ABC$, the altitudes from the vertices $A,B,C$ opposite sides are in H.P., then $\sin A,\sin B,\sin C$ are in

• A. H.P.
• B. Arithmetic-Geometric progression
• C. A.P.
• D. G.P.

Answer 1

Answer: (C)

A.P.

Altitude from $A$ to $BC$ is $AD$

Area of $△=\frac{1}{2}AD\times BC$

$\therefore \frac{2Areaof△}{a}=AD$

Similarly, $\frac{2Areaof△}{b}=BE$ and

$\frac{2Areaof△}{c}=CF$

Given that, $AD,BE,CF$ are in H.P.

$\therefore \frac{2△}{a},\frac{2△}{b},\frac{2△}{c}$ are in H.P.

$\Rightarrow \frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in H.P.

$\Rightarrow a,b,c$ are in A.P.

Therefore, $\sin A,\sin B,\sin C$ are in A.P ....[ From SIne rule ]