Question

If in a $△PQR,\sin P,\sin Q,\sin R$ are in $AP$, then

• 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 $HP$

By sine rule
$\frac{a}{\sin P}=\frac{b}{\sin Q}=\frac{c}{\sin R}=k$
Also, $\frac{1}{2}a{p}_1=△$
$\Rightarrow \frac{2△}{a}=p_1$
$\Rightarrow p_1=\frac{2△}{k\sin P}$
Similarly, $p_1=\frac{2△}{k\sin Q}$ and $p_3=\frac{2△}{k\sin R}$
Since, $\sin P,\sin Q,\sin R$ are in $AP$.
Therefore, $\frac{1}{\sin P}\cdot \frac{1}{\sin Q},\frac{1}{\sin R}$ are in $HP$
$\Rightarrow \frac{2△}{k\sin P},\frac{2△}{k\sin Q},\frac{2△}{k\sin R}$ are in $HP$
$\Rightarrow p_1,p_2,p_3$ are in $HP$.