Two corners of a triangle have angles of #pi / 3 # and # pi / 2 #. If one side of the triangle has a length of #7 #, what is the longest possible perimeter of the triangle?

1 Answer
May 12, 2016

Longest possible perimeter is #33.124#.

Explanation:

As two angles are #pi/2# and #pi/3#, the third angle is #pi-pi/2-pi/3=pi/6#.

This is the least angle and hence side opposite this is smallest.

As we have to find longest possible perimeter, whose one side is #7#, this side must be opposite the smallest angle i.e. #pi/6#. Let other two sides be #a# and #b#.

Hence using sine formula #7/sin(pi/6)=a/sin(pi/2)=b/sin(pi/3)#

or #7/(1/2)=a/1=b/(sqrt3/2)# or #14=a=2b/sqrt3#

Hence #a=14# and #b=14xxsqrt3/2=7xx1.732=12.124#

Hence, longest possible perimeter is #7+14+12.124=33.124#