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

Engineering

Mathematics

Circumradius Inradius and exradius

Question

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

The area of the circumcircle of the triangle PQR is 100π.

$∠$ QPR = 45°.

The radius of the incircle of the triangle PQR is 10 $\sqrt{3}$ – 15.

The area of the triangle PQR is 25 $\sqrt{3}$ and $∠$ QRP = 120°.

LiveFree JEE Main Previous Year Online Papers Live Free JEE Advance Previous Year Online Papers

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

$cos 3 0 º = \frac{10^{2} + {(10 \sqrt{3})}^{2} - x^{2}}{2 \cdot 10 \cdot 10 \sqrt{3}} \frac{\sqrt{3}}{2}$

⇒ x = 10 = PR

$∠$ PRQ = 120º

Ar ( $∆$ PQR) = $\frac{1}{2}$ (PR) (QR) × sin 120º = $25 \sqrt{3}$

$r = \frac{Δ}{S} = \frac{25 \sqrt{3}}{10 + 5 \sqrt{3}} = \frac{5 \sqrt{3}}{2 + \sqrt{3}} = 5 \sqrt{3} (2 - \sqrt{3})$

$R = \frac{PQ}{2 \sin 120 º} = 10$