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Question

In a triangle XYZ, let x, y, z, be the lengths of sides opposite to the angles X, Y, Z, respectively, and 2s=x+y+z. If sx4=sy3=sz2 and the area of the incircle of the triangle XYZ is 8π3, then:

A
Area of the triangle XYZ is 66
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B
The radius of the circumcircle of the triangle XYZ is 3566
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C
sinX2sinY2sinZ2=435
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D
sin2(X+Y2)=35
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Solution

The correct option is D sin2(X+Y2)=35
Let sx4=sy3=sz2=k
sx=4k, sy=3k, sz=2k
s=9k (add the above equations)
x=5k, y=6k, z=7k

Area of the incircle =8π3
r2=83
Δ2s2=83 (r=Δs)
k=1 (put Δ=s(sx)(sy)(sz) )

s=9, x=5, y=6, z=7
Δ=s(sx)(sy)(sz)
Δ=66

R=xyz4Δ
R=3546

r=4RsinX2sinY2sinZ2
sinX2sinY2sinZ2=435

sin2(X+Y2)=cos2Z2=s(sz)xy=35

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