In `trianglePQR, angleP : angleQ : angleR= 1:2:3`. Prove that `p:q:r = 1:sqrt(3):2`

Explanation

Let the common ratio be`x`.
`therefore` Angles are `x,2x` and `3x`
Now, `x+2x+3x=180^(@)` [Angle sum property of a triangle]
`6x=180^(@)`
`x=30^(@)`
`therefore` Angles are `30^(@), 60^(@)` and `90^(@)`
Now, by sine rule,
`p/(sinP) = q/(sinQ)= r/(sinR)`
`rArr p/(sin 30^(@)) = q/(sin60^(@))= r/(sin90^(@))`
`p/(1/2) = q/(sqrt(3)/2) = r/1=k` (say)
`p:q:r= k/2 : (sqrt(3)k)/(2) : k`
`therefore` The required ratio is `1:sqrt(3):2`
Hence proved.