A triangle has two corners with angles of # pi / 12 # and # (7 pi )/ 8 #. If one side of the triangle has a length of #13 #, what is the largest possible area of the triangle?

Question

A triangle has two corners with angles of #  pi  /  12  # and # (7 pi  )/  8   #.  If one side of the triangle has a length of #13  #, what is the largest possible area of the triangle?

Theodore Abad · Accepted Answer

Largest possible area of the triangle is #color(blue)(A_t = 64.1194)#

Given #/_A = (7pi)/8, /_B = pi /12#

#/_C = pi - (7pi)/8 - pi/12 = pi / 24#

Smallest angle is #/_C = pi/24#

To get the largest area of the triangle possible, smallest angle should correspond to the given length 13.

#i.e c = 13#

We know,
#a / sin A = b / sin B = c / sin C#

Hence,
#a / sin ((7pi)/8) = b / sin (pi/12) = 13 / sin (pi / 24)#

#a = (13 * sin ((7pi)/8)) / sin (pi/24) = 38.1141#

#b = (13 * sin (pi/12)) / sin (pi / 24) = 25.7776#

Area of triangle #A_t # =1/2 . Base . Height

Base #a = 38.1141#

Height #h = c sin (/_B) = 13 * sin (pi / 12) = 3.3646#

#A_t = (1/2) * 38.1141 * 3.3646 = color(blue)(64.1194)#

Largest possible area of the triangle is #color(blue)(A_t = 64.1194)#

Verification :

#**A_t** = (1/2) * a * b sin C = (1/2) * 38.1141 * 25.7776 * sin (pi/24) =**64.1194**#

Maverick Smith · Answer

undefined The largest possible area of the triangle can be found using the formula:

Area = 1/2 * a * b * sin(C)

Where 'a' and 'b' are the lengths of the two sides forming the given angles, and C is the angle between these sides.

Given that one side of the triangle has a length of 13 and the two given angles are π/12 and (7π)/8, we can use the Law of Sines to find the lengths of the other two sides.

Let's denote the length of the other side corresponding to the angle π/12 as 'x':

sin(π/12) / 13 = sin((7π)/8) / x

From this, we can solve for 'x' to find the length of the third side.

Then, we can calculate the area of the triangle using the formula mentioned earlier. The largest possible area will be obtained when the angle between the sides is at its maximum value, which is π/2 (90 degrees).

Therefore, the largest possible area of the triangle can be calculated using the given information and the formula for the area of a triangle.