The angle of a right angled triangle are A.P. The ratio of the in-radius and the perimeter is

Let ABC be the right angled triangle whose anglesare in A.P. Then,Now, A+B+C=180∘⇒3B=180∘⇒B=60∘So, let the angles be A=30∘,B=60∘ and C=90∘∴ asin⁡30∘=bsin⁡60∘=csin⁡90∘=2R⇒ a=R,b=3R and c=2RAlso,Δ=12absin⁡90∘=12ab=32R2∴ rs=Δs2⇒ rs=3R223R+2R)2−32×4(22=233+33+32⇒ rs=23(3−3)2(9−3)263(3−1)236=23(4−23)⇒ rs=2−33−⇒r2s=2−323