Question

Let $△ABC,x,y,z$ are the length of altitudes drawn from $A,B,C$ respectively.

If $x,y,z$ are in $A.P.$ then?

• A. $\cos A,\cos B,\cos C$ are in $A.P.$
• B. $\cos A,\cos B,\cos C$ are in $H.P.$
• C. $\sin A,\sin B,\sin C$ are in $H.P.$
• D. $\sin A,\sin B,\sin C$ are in $A.P.$

Answer 1

The correct option is C $\sin A,\sin B,\sin C$ are in $H.P.$

It is given that $x,y,z$ are the length of altitudes drawn from $A,B,C$

Thus the area of the triangle is

$\Delta =\frac{1}{2}ax\Rightarrow x=\frac{2\Delta }{a}=\frac{2\Delta }{2R\sin A}=\frac{\Delta }{R\sin A}$

Similarly, $y=\frac{\Delta }{R\sin B},z=\frac{\Delta }{R\sin C}$

Given $x,y,z$ are in $A.P.$

$\Rightarrow \frac{\Delta }{R\sin A},\frac{\Delta }{R\sin B},\frac{\Delta }{R\sin C}$ are in $A.P.$

$\Rightarrow \sin A,\sin B,\sin C$ are in $H.P.$