In a non right angled triangle `DeltaPQR`, let `p,q,r` denote the lengths of the sides opposite to the angle P,Q,R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. if `p=sqrt(3),q=1` and the radius of the circumcircle of the `DeltaPQR` equals to 1, then which of the followign options is/are correct?

Options

A. length of `RS=(sqrt(7))/(2)`
B. length of `OE=(1)/(6)`
C. Radius of incircle of `DeltaPQR=(sqrt(3))/(2)(2-sqrt(3))`
D. Area of `DeltaSOE=(sqrt(3))/(12)`

Answer

A, B, C

Explanation

`(p)/(sinP)=(q)/(sinQ)=2(1)impliessinP=(sqrt(3))/(2),sinQ=(1)/(2)`
`impliesangleP=60^(@)` or `120^(@)andangleQ=30^(@)or 150^(@)`
because `angleP+angleQ` must be less than `180^(@)` but not equal to `90^(@)`
`angleP=120^(@)and angle P=30^(@)` and `angleR=30^(@)" "(r)/(sinR)=2impliesr=1`
Noe length of median `RS=(1)/(2)sqrt(2p^(2)+2q^(2)-r^(2))=(1)/(2)sqrt(6+2-1)=(sqrt(7))/(2)implies` option (A) is correct
Irradius `=(2Delta)/(p+q+r)=((2pqr)/(4xx(1)))/(p+q+r)=(1)/(2)((1xx1xxsqrt(3))/(1+1+sqrt(3)))=(sqrt(3))/(2)((2-sqrt(3))/(1))implies` options (C) is correct
`impliesOE=(2("area of "DeltaOQR))/(QR)=(2xx(1)/(3)((1)/(2).1.sqrt(3)sin30^(@)))/(sqrt(3))=(1)/(6)`