Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively.

If $\sin A,\sin B$ and $\sin C$ are in AP then which of the following is true?

• A. the altitudes are in AP
• B. the altitudes are in HP
• C. the medians are in GP
• D. the medians are in AP

Answer 1

The correct option is B the altitudes are in HP

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

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

$\Delta =\frac{1}{2}a{p}_1,\Delta =\frac{1}{2}b{p}_2,\Delta =\frac{1}{2}c{p}_3{p}_1=\frac{2\Delta }{a},p_2=\frac{2\Delta }{b},p_3=\frac{2\Delta }{c}$

Now $a,b,c$ are in $A.P$

So the altitudes are in $H.P$