Question

If the angles of the triangle are in the ratio 3:4:5, then how do you find the ratio of the sides?

Answer 1

(Assuming I haven't messed up)
The ratio of the sides (to 4 significant digits) is

`1 : 1.225 : 1.366`

Explanation:

Part 1
For any triangle with sides `a, b, c` and corresponding opposite angles `A, B, C` the Law of Sines tells us that

`(sin(A))/a = (sin(B))/b=(sin(C))/c`

or, by rearranging:

`b/a = (sin(B))/(sin(A))`

and

`céa=sin(C)/sin(A)`

Part 2
If `/_A:/_B:/_C` are in the ratio `3:4:5`
since `/_A+/_B+/_C = pi`

`/_A = (3pi)/12 = pi/4`

`/_B=(4pi)/12= pi/3`

`/_C= (5pi)/12`

Part 3
`a:b:c`
`color(white)("XXX")= a/a:b/a:c/a`

`color(white)("XXX")=1 : sin(B)/sin(A) : sin(C)/sin(A)`

`color(white)("XXX")=1 : sin(pi/3)/sin(pi/4) : sin((5pi)/12)/sin(pi/4)`

(and using my calculator):
`color(white)("XXX")= 1 :1.224745 : 1.366025`