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

1 Answer
Dec 5, 2017

#Area of # largest possible #Delta = color (purple)(42.44)#

Explanation:

Three angles are #pi/2, (pi/6, (pi -( (pi/2)+ ((pi)/6)) =( pi/3#

#a/ sin A = b / sin B = c / sin C#

To get the largest possible are, smallest angle should correspond to the side of length 7

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

#b = (7*sin (pi/2)) / sin (pi / 6) = (7*1) / (1/2) = 14#

#c = (7* sin (pi/3)) / sin (pi/6) = (7 * (sqrt3/2))/(1/2) = 7 sqrt 3 = 12.1245#

Semi perimeter #s = (a + b + c) / 2 = (7 + 14 + 12.1245)/2 = 16.5623#

#s-a = 16.5623-7 = 9.5623#
#s-b = 16.5623-14 = 2.5623#
#s-c = 16.5623 - 12.1245 = 4.4378#

#Area of Delta = sqrt (s (s-a) (s-b) (s-c))#

#Area of Delta = sqrt( 16.5623 * 9.5623 * 2.5623 * 4.4378)#
#Area of# largest possible #Delta = color (purple)(42.44)#

Alternate method to calculate #Area of Delta#
Since it’s a right triangle,
#Area of Delta = (1/2) Base * ht = (1/2) * 7 * 12.1245 = color (purple)(42.44)#