Question

In $△ABC$ the sides opposite to angles $A,B,C$ are $a,b,c$ respectively. If $\sin A,\sin B,\sin C$ are in $A.P.,$ Then which of the following is true ?

• A. the altitudes are in $A.P.$
• B. the altitudes are in $G.P.$
• C. the altitudes are in $H.P.$
• D. the altitudes are in $A.G.P.$

Answer 1

The correct option is C the altitudes are in $H.P.$

Using sine rule :
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$

As, $\sin A,\sin B,\sin C$ are in $A.P.$
$\Rightarrow a,b,c$ are also in $A.P.$
Now, area of triangle is :
$\Delta =\frac{1}{2}a{p}_1=\frac{1}{2}b{p}_2=\frac{1}{2}c{p}_3$
Here, $p_1,p_2,p_3$ are altitudes from $A,B,C$ respectively.
$\Rightarrow p_1=\frac{2\Delta }{a},p_2=\frac{2\Delta }{b},p_3=\frac{2\Delta }{c}$
As, $a,b,c$ are in $A.P.$
$\therefore p_1,p_2,p_3$ are in $H.P.$