Question

If in a triangle $PQR,\sin P,\sin Q,\sin R$ 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. Nothing can be said.

Answer 1

The correct option is B The altitudes are in $H.P.$

Let $a,b,c$ be the altitudes of the respective sides $QR,RP,PQ$ of the triangle.
$\therefore \Delta =\frac{pa}{2}=\frac{qb}{2}=\frac{rc}{2}$
$\Rightarrow a=\frac{2\Delta }{p},b=\frac{2\Delta }{q},c=\frac{2\Delta }{r}$
$\Rightarrow a=\frac{2\Delta }{K\sin P},b=\frac{2\Delta }{K\sin Q},c=\frac{2\Delta }{K\sin R}$
Given, $\sin P,\sin Q$ and $\sin R$ are in $A.P.$
So the terms containing its reciprocals must be in $H.P.$
$\Rightarrow a,b,c$ are in $H.P.$
$\Rightarrow$Altitudes of $\Delta PQR$ are in $H.P.$