Let 'P' be an interior point of `Delta ABC`. If `angle A=45^(@), angle B=60^(@)` and `angle C=75^(@)`. If X=area of `Delta PBC,Y=` area of `Delta PAC` and Z = area of `Delta PAB`, then which of the following ratios is/are true ?

Options

A. If P is the centroid, then X : Y : Z is 1 : 1 : 1
B. If P is the incentre, then X : Y : Z is `2 :sqrt(6) :(sqrt(3)+1)`
C. If P the orthocentre, then X : Y : Z is `1 : sqrt(3) :(2+sqrt(3))`
D. If P is the circumcentre, then X : Y : Z is `2 : sqrt(3) :1`

Answer

A, B, C, D

Explanation

(a)
Using properties of median, we have
`Delta PBC = Delta PCA = Delta PAB`
`therefore Delta PBC : Delta PCA : Delta PAB = 1:1:1`
(b)
`Delta PBC : Delta PCA : Delta PAB`
`=(1)/(2)ar : (1)/(2)br : (1)/(2)cr`
`= a:b:c`
`= sin 45^(@): sin 60^(@): sin 75^(@)=2 : sqrt(6):(sqrt(3)+1)`
(c )
`Delta PBC : Delta PCA : Delta PAB`
`=(1)/(2)a(2R cos B cos C) : (1)/(2)b(2 R cos C cos A) : (1)/(2)c(2R cos A cos B)`
= sin A cos B cos C : sin B cos C cos A : sin C cos A cos B
`= tan 45^(@) : tan 60^(@) : tan 75^(@)`
`= 1: sqrt(3):(2+sqrt(3))`
(d)
`Delta PBC : Delta PCA : Delta PAB`
`=(1)/(2)R^(2)sin 2A : (1)/(2)R^(2)sin 2B(1)/(2)R^(2)sin 2C`
`= sin 2A : sin 2B : sin 2C`
`= sin 90^(@) : sin 120^(@) : sin 150^(@)`
`= 2 : sqrt(3):1`