JEE Advanced2018MathematicsProperties of TrianglesActual
In a triangle P Q R , l e t ∠ P Q R = 30 o and the sides P Q a n d Q R have lengths 10 3 a n d 10 , respectively. Then, which of the following statement(s) is (are) TRUE?
Options
- A∠ Q P R = 45 o
- BThe area of the triangle P Q R i s 25 3 a n d ∠ Q R P = 120 o
- CThe radius of the in circle of the triangle P Q R i s 10 3 - 15
- DThe area of the circumcircle of the triangle P Q R i s 100 π
Correct answer
B. The area of the triangle P Q R i s 25 3 a n d ∠ Q R P = 120 o
Step-by-step solution
cos 30 o = 10 3 2 + 10 2 - P R 2 2 × 10 3 × 10 ⇒ P R = 10 ∵ Q R = P R ⇒ ∠ P Q R = ∠ Q P R ∠ Q P R = 30 o B Area of ∆ P Q R = 1 2 × 10 3 × 10 × sin 30 o = 1 2 × 10 × 10 3 × 1 2 = 25 3 ∠ Q R P = 180 o - 30 o + 30 o = 120 o C r = ∆ S = 25 3 10 + 10 + 10 3 2 = 25 3 10 + 5 3 = 5 3 . 2 - 3 = 10 3 - 15 D R = a 2 sin A = 10 2 sin 30 o = 10 ∴ A r e a = π R 2 = 100 π