Question

If in a triangle $ABC,\sin A,\sin B,\sin C$ are in A.P., then

• A. the altitudes are in A.P.
• B. the altitudes are in H.P.
• C. the altitudes are in G.P.
• D. none of these

Answer 1

Answer: (B)

the altitudes are in H.P.

If $p_1,p_2,p_3$, are the altitudes from A, B, C respectively,
then $\Delta =\frac{1}{2}a{p}_2=\frac{1}{2}b{p}_2=\frac{1}{2}c{p}_3$
$\Rightarrow p_1=\frac{2\Delta }{a},p_2=\frac{2\Delta }{b},p_3=\frac{2\Delta }{c}$
By the law of sines
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k\left(say\right)$
$\therefore =\frac{2\Delta }{k\sin A},p_2=\frac{2\Delta }{k\sin B},p_3=\frac{2\Delta }{k\sin C}$
Now, $\sin A,\sin B,\sin C$ are in A.P.
$\Rightarrow p_1,p_2,p_3$ are in H.P.