In a triangle PQR, let `anglePQR = 30^(@)` and the sides PQ and QR have lengths `10 sqrt3` and 10, respectively. Then, which of the following statement(s) is (are) TRUE ?

Options

A. `angleQPR = 45^(@)`
B. The area of the triangle PQR is `25sqrt3 and angleQRP = 120^(@)`
C. The radius of the incircle of the triangle PQR is `10 sqrt3 -15`
D. The area of the circumcircle of the triangle PQR is `100 pi`

Answer

B, C, D

Explanation

`cos Q (100 + 300 - (PR)^(2))/(2 xx 10 xx 10 sqrt3)`
`rArr cos 30^(@) = (400 -(PR)^(2))/(200 sqrt3)`
`rArr 300 = 400 - (PR)^(2)`
`rArr PR = 10`
So, triangle is isosceles.
`:. angleQPR = 30^(@) and anglePRQ = 120^(@)`
Area of triangle, `Delta = (1)/(2) xx (PQ) xx (QR) sin Q`
`= (1)/(2) xx 10 xx 10 sqrt3 xx (1)/(2) = 25 sqrt3`
Now, `2s = 20 + 10 sqrt3`
Inradius, `r = (Delta)/(s) = (25 sqrt3)/(10 + 5 sqrt3) = (5sqrt3)/(2+sqrt3)`
`= (5 sqrt3)/(2 + sqrt3) xx (2 - sqrt3)/(2 - sqrt3)`
`= 10 sqrt3 - 15`
Using sine rule, we get
`(PR)/(sin Q) = 2R'`, where R' is circumradius
or `(10)/(sin 30^(@)) = 2R'`
`:. R' = 10`
Hence, area of circumcircle `= pi (R')^(2) = 100 pi`