A triangle has two corners with angles of # (2 pi ) / 3 # and # ( pi )/ 6 #. If one side of the triangle has a length of #7 #, what is the largest possible area of the triangle?

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Answer 1 · 2018-01-20T14:47:41

Largest possible triangle will have sides #color(brown)(7, 7, 12.1244)#

Smallest possible triangle will have sides #color(red)(7, 4.0415, 4.0415)#

The three angles are #(2pi)/3, pi/6, pi/6# as sumof the three angles equals #pi^c#. It’s an isosceles triangle as two angles measure #pi/6# each.

Case 1 : To get largest triangle possible

Side 7 should correspond to the smallest angle #pi/6#

#:.7 / sin (pi/6) = a / sin ((2pi)/3)#

#a = (7 * sin ((2pi)/3)) / sin (pi/6) ~~ 12.1244#

Three sides are #color(brown)(7, 7, 12.1244)#

Case 2 : To get smallest triangle possible

Side 7 should correspond to the largest angle #((2pi)/3)#

#7 / sin ((2pi)/3) = b / sin (pi/6)#

#b = (7 * sin (pi/6)) / sin ((2pi)/3) = 4.0415#

Three sides are #color(red)(7, 4.0415, 4.0415)#