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Question:

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

∠QPR = 45°

The area of the triangle PQR is 25√3 and ∠QRP = 120°

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

The radius of the incircle of the triangle PQR is 10√3/5

Solution:

The correct options are B, C, and D.
(A) cos 30° = (10√3)² + (10)² - (PR)² / (2 × 10√3 × 10) ⇒ √3/2 = 400 - (PR)² / 200√3 ⇒ PR = 10. Since QR = PR, ∠PQR = ∠QPR. ∠QPR = 30°
(B) Area of ΔPQR = 1/2 × 10√3 × 10 × sin 30° = 1/2 × 10 × 10√3 × 1/2 = 25√3. ∠QRP = 180° - (30° + 30°) = 120°
(C) r = Δ/S = 25√3 / (10 + 10 + 10√3/2) = 25√3 / (10 + 5√3) = 5√3 × (2 - √3) = 10√3/5
(D) R = a / 2sinA = 10 / 2sin30° = 10. Therefore, Area = πR² = 100π