Question

In a triangle $PQR$, the median of side $QR$ is of length $\sqrt{\frac{7+5\sqrt{3}}{13}}$ and it divides $\angle P$ in to the angles of ${30}^0$ and ${45}^0$. What is the length of side $QR$?

Answer 1

To find: Length of QR

By $m-ncot$ theorem

$2\cot \theta =\cot 30-\cot 45$

$\cot \theta =\frac{\sqrt{3}-1}{2}$

$\sin R=\sin (180-45-\theta )$

$\sin R=\sin (45+\theta )$

$\sin R=\frac{1}{\sqrt{2}}\sin \theta +\cos \theta$

$\sin R=\frac{1}{\sqrt{2}}\frac{\sqrt{3}+1}{\sqrt{8-2\sqrt{3}}}$

Apply sine rule in $△PKR$

$\frac{PK}{\sin R}=\frac{KR}{\sin 45}=\frac{\frac{QR}{2}}{\sin 45}$

$QR=\frac{\sqrt{2}\times \sqrt{7+5\sqrt{3}}}{\sqrt{13}}\times \sqrt{2}\times \frac{\sqrt{8-2\sqrt{3}}}{\sqrt{3}+1}$

$QR=\frac{2}{\sqrt{3}+1}\frac{\sqrt{7+5\sqrt{3}}\sqrt{8-2\sqrt{3}}}{\sqrt{13}}$

$QR=\frac{2}{\sqrt{3}+1}\frac{\sqrt{26+26\sqrt{3}}}{\sqrt{13}}$

$QR=\frac{2}{\sqrt{3}+1}\sqrt{2(\sqrt{3}+1}$

QR$=\frac{2\sqrt{2}}{\sqrt{3}+1}$