Properties of Triangle Given Angle and Two Adjacent Side Lengths

To determine which statement(s) are true for the triangle $PQR$, let us denote the sides opposite to the vertices $P, Q,$ and $R$ as $p, q,$ and $r$ (or using the given notation, $a = QR = 10$, $b = PR$, and $c = PQ = 10\sqrt{3}$).

Step 1: Finding the third side $PR$ and angles

Using the Law of Cosines for $\triangle PQR$ at vertex $Q$: $PR^2 = PQ^2 + QR^2 - 2(PQ)(QR)\cos(\angle PQR)$

Substituting the given values $\angle PQR = 30^\circ$, $PQ = 10\sqrt{3}$, and $QR = 10$: $PR^2 = (10\sqrt{3})^2 + 10^2 - 2(10\sqrt{3})(10)\cos 30^\circ$ $PR^2 = 300 + 100 - 200\sqrt{3} \left(\frac{\sqrt{3}}{2}\right)$ $PR^2 = 400 - 300 = 100 \implies PR = 10$

Since $PR = QR = 10$, the triangle $PQR$ is isosceles with $\angle QPR = \angle PQR$. $\angle QPR = 30^\circ$ The third angle $\angle QRP$ is: $\angle QRP = 180^\circ - (30^\circ + 30^\circ) = 120^\circ$

• Option (A): $\angle QPR = 45^\circ$ is FALSE (as $\angle QPR = 30^\circ$).

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Step 2: Area of $\triangle PQR$

The area ($\Delta$) of $\triangle PQR$ is given by: $\Delta = \frac{1}{2} \cdot PQ \cdot QR \cdot \sin(\angle PQR)$ $\Delta = \frac{1}{2} \cdot (10\sqrt{3}) \cdot (10) \cdot \sin 30^\circ = 50\sqrt{3} \cdot \frac{1}{2} = 25\sqrt{3}$

Since the area is $25\sqrt{3}$ and $\angle QRP = 120^\circ$:

• Option (B): is TRUE.

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Step 3: Inradius ($r$) of $\triangle PQR$

The semi-perimeter $s$ of the triangle is: $s = \frac{PQ + QR + PR}{2} = \frac{10\sqrt{3} + 10 + 10}{2} = 10 + 5\sqrt{3}$

The inradius $r$ is given by: $r = \frac{\Delta}{s} = \frac{25\sqrt{3}}{10 + 5\sqrt{3}} = \frac{5\sqrt{3}}{2 + \sqrt{3}}$

Rationalizing the denominator: $r = 5\sqrt{3}(2 - \sqrt{3}) = 10\sqrt{3} - 15$

• Option (C): is TRUE.

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Step 4: Circumradius ($R$) and Area of Circumcircle

Using the Sine Rule: $2R = \frac{PR}{\sin(\angle PQR)} = \frac{10}{\sin 30^\circ} = \frac{10}{1/2} = 20 \implies R = 10$

The area of the circumcircle is: $\text{Area} = \pi R^2 = \pi (10)^2 = 100\pi$

• Option (D): is TRUE.

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Conclusion:

The correct statements are B, C, and D.