Question

In a non-right-angle triangle $\Delta PQR$, let $p,q,r$ denote the lengths of the sides opposite to the angles at $P,Q,R$ respectively. The median from $R$ meets the side $PQ$ at $S$, the perpendicular from $P$ meets the side $QR$ at $E$, and $RS$ and $PE$ intersect at $0$. If $p=\sqrt{3},q=1$, and the radius of the circumcircle of the $\Delta PQR$ equals $1$, then which of the following options is/are correct?

• A. Area of $\Delta SOE=\sqrt{3}/12$
• B. Radius of incircle of $\Delta PQR=\frac{\sqrt{3}}{2}[2-\sqrt{3}]$
• C. Length of $RS=\frac{\sqrt{7}}{2}$
• D. Length of $OE=\frac{1}{6}$

Answer 1

Answer: (B), (C), (D)

Length of $OE=\frac{1}{6}$

Explanation for the correct options.

Step 1: Find the value of $r$.

Using sine law,

$\frac{p}{\sin P}=\frac{q}{\sin Q}=\frac{r}{\sin R}=2R$ (where $R$ is the radius of the circumcircle of the $\Delta PQR$).

$\Rightarrow \frac{\sqrt{3}}{\sin P}=\frac{1}{\sin Q}=\frac{r}{\sin R}=2\left(1\right)$

$\begin{array}{rcl}& \Rightarrow & \sin P=\frac{\sqrt{3}}{2},\sin Q=\frac{1}{2}\\ \Rightarrow \angle P& =& 60°\mathrm{or}120°\\ \Rightarrow \angle Q& =& 30°\mathrm{or}150°\end{array}$

We know that, $\angle P+\angle Q<180°and\angle P+\angle Q\ne 90°$, so $\angle P=120°,\angle Q=30°\mathrm{and}\angle R=30°$.

So,

$$\begin{gathered} \frac{r}{2}=\frac{1}{2} \\ \Rightarrow r=1 \end{gathered}$$

Step 2: Find the length of $RS$.

Using the formula of length of median, we get

$\begin{array}{rcl}RS& =& \frac{1}{2}\sqrt{2p^2+2q^2-r^2}\\ & =& \frac{1}{2}\sqrt{6+2-1}\\ & =& \frac{\sqrt{7}}{2}\end{array}$

Hence, option C is correct.

Step 3: Find the length of $OE$.

$\begin{array}{rcl}OE& =& \frac{1}{3}PE\\ & =& \frac{1}{3}\times \frac{1}{2}\sqrt{2r^2+2q^2-p^2}\\ & =& \frac{1}{6}\sqrt{2+2-3}\\ & =& \frac{1}{6}\end{array}$

Hence, option D is correct.

Step 4: Find the radius of incircle of $\Delta PQR$.

$\begin{array}{rcl}& =& \frac{2∆}{p+q+r}\\ & =& \frac{{\displaystyle \frac{2pqr}{4\left(1\right)}}}{2+\sqrt{3}}\\ & =& \frac{{\displaystyle \frac{\sqrt{3}}{2}}}{2+\sqrt{3}}\\ & =& \frac{\sqrt{3}}{2}\left(2-\sqrt{3}\right)\end{array}$

Hence, option B is correct.

Explanation for the incorrect option.

Option A

$\begin{array}{rcl}ar\left(\Delta SOE\right)& =& \frac{1}{3}\mathrm{ar}\left(\Delta SER\right)=\frac{1}{12}\mathrm{ar}\left(\mathrm{\Delta PQR}\right)\left[\mathrm{by}\mathrm{ar}\left(\mathrm{\Delta SE}R\right)=\frac{1}{4}\mathrm{ar}\left(\mathrm{\Delta PQR}\right)\right]\\ & =& \frac{1}{12}\times \frac{\sqrt{3}}{4}\\ & =& \frac{\sqrt{3}}{48}\end{array}$

Hence, option A is incorrect.

Hence, option B, C, and D are correct.