Two corners of a triangle have angles of # ( pi )/ 3 # and # ( pi ) / 4 #. If one side of the triangle has a length of # 9 #, what is the longest possible perimeter of the triangle?

1 Answer
May 24, 2016

Perimeter is #32.314#

Explanation:

As two angles of a triangle are #pi/3# and #pi/4#, the third angle is

#pi-pi/3-pi/4=(12-4-3)pi/12=(5pi)/12#

Now for the longest possible perimeter, the given side say #BC#, should be the smallest angle #pi/4#, let this be #/_A#. Now using sine formula

#9/sin(pi/4)=(AB)/sin(pi/3)=(AC)/sin((5pi)/12)#

Hence #AB=9xxsin(pi/3)/sin(pi/4)=9xx(sqrt3/2)/(sqrt2/2)=9xx1.732/1.414=11.02#

and #AC=9xxsin((5pi)/12)/sin(pi/4)=9xx0.9659/(1.4142/2)=12.294#

Hence, perimeter is #9+11.02+12.294=32.314#