amths~

A:B:C = 1:2:3
Let A = x, B = 2x and C = 3x.
x + 2x + 3x = 180度 (angle sum of triangle)
x = 30度
So A = 30度, B = 2(30)度 = 60度 and C = 3(30)度 = 90度
By Sine Formula, we have
a/sin 30度 = b/sin 60度 = c/sin 90度
a/(1/2) = b/[sq. rt. (3)/2] = c/1
a/1 = b/[sq. rt. (3)] = c/2
So a:b:c = 1:[sq. rt. (3)]:2

Since A:B:C = 1:2:3, B = 2A and C = 3A.
In addition, A + B + C = 180
=> A + 2A + 3A = 180 or A = 30, B = 60, C = 90
By sine rule, a / sin A = b / sin B = c / sin C = k where k is a constant
=> a = k sin A, b = k sin B and c = k sin C
=> a : b : c =sin A : sin B : sin C
= sin 30 : sin 60 : sin 90
= (1/2) : (sqrt(3)/2) : 1
= 1 : sqrt(3) : 2