Two corners of a triangle have angles of # ( pi )/ 3 # and # ( pi ) / 6 #. If one side of the triangle has a length of # 5 #, what is the longest possible perimeter of the triangle?

Question

Two corners of a triangle have angles of #  (  pi  )/  3  # and #  ( pi ) /  6  #. If one side of the triangle has a length of # 5  #, what is the longest possible perimeter of the triangle?

Claire Akin · Accepted Answer

#=11.83# Clearly this is a right angled triangle as #pi-(pi)/3-pi/6=pi/2# One #side=hypoten use =5# ;So other sides #=5sin(pi/3) and 5cos( pi/3)# Therefore Perimeter of the triangle#=5+5sin(pi/3)+ 5cos( pi/3)# #=5+(5times0.866)+(5times0.5)# #=5+4.33+2.5)# #=11.83#

Caroline Aucoin · Answer

undefined To find the longest possible perimeter of the triangle given two angles and one side length:

1. Determine the measure of the third angle by subtracting the sum of the given angles from $ \pi $.
2. Use the Law of Sines to find the lengths of the other two sides.
3. Calculate the perimeter by adding the lengths of all three sides.
4. Choose the maximum value for the perimeter.

The longest possible perimeter of the triangle will be obtained when the third angle is as small as possible, making the triangle as close to equilateral as possible. Therefore, the third angle will be $ \frac{\pi}{2} $. Then apply the Law of Sines to find the lengths of the other two sides, and calculate the perimeter using the given side length.