Question

In a triangle $PQR$, let $\angle PQR=30^{\circ }$ and the sides $PQ$ and $QR$ have lengths $10\sqrt{3}$ and 10, respectively. Then, which of the following statement(s) is (are) TRUE?

• A. $\angle QPR=45^{\circ }$
• B. The area of the triangle $PQR$ is $25\sqrt{3}$ and $\angle QPR=120^{\circ }$
• C. The radius of the incircle of the triangle $PQR$ is $10\sqrt{3}-15$
• D. The area of the circumcircle of the triangle $PQR$ is $100\pi$

Answer 1

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

$\cos 30^{\circ }=\frac{{\left(10\sqrt{3}\right)}^2+10^2-P{R}^2}{2\left(10\sqrt{3}\right)\left(10\right)}$
$\Rightarrow PR=10$
$\angle QPR=30^{\circ }$
Area of triangle PQR $=\frac{1}{2}(10)(10\sqrt{3})(\sin 30^{\circ })$
$=25\sqrt{3}$
Also, $r=\frac{\Delta }{s}=\frac{25\sqrt{3}}{10+5\sqrt{3}}$
$=5\sqrt{3}(2-\sqrt{3})=10\sqrt{3}-15$
$R=\frac{abc}{4\Delta }=\frac{(10)(10)(10\sqrt{3})}{4(25\sqrt{3})}=10$
Area of the circumcircle $=\pi (10{)}^2=100\pi$