Two corners of a triangle have angles of # (5 pi )/ 8 # and # ( pi ) / 12 #. If one side of the triangle has a length of # 18 #, what is the longest possible perimeter of the triangle?

1 Answer
Oct 22, 2016

Longest possible perimeter is #137.434#

Explanation:

As two angles are #(5pi)/8# and #pi/12#, third angle is

#pi-(5pi)/8-pi/12=(24pi)/24-(15pi)/24-(2pi)/24=(7pi)/24#

the smallest of these angles is #pi/12#

Hence, for longest possible perimeter of the triangle, the side with length #18#, will be opposite the angle #pi/12#.

Now for other two sides, say #b# and #c#, we can use sine formula, and using it

#18/sin(pi/12)=b/sin((5pi)/8)=c/sin((7pi)/24)#

or #18/0.2588=b/0.9239=c/0.7933#

therefore #b=(18xx0.9239)/0.2588=64.259#

and #c=(18xx0.7933)/0.2588=55.175#

and perimeter is #64.259+55.175+18=137.434#