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

1 Answer
Sep 23, 2015

Answer:

(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#