A triangle has sides A, B, and C. Sides A and B have lengths of 10 and 8, respectively. The angle between A and C is #(11pi)/24# and the angle between B and C is # (3pi)/8#. What is the area of the triangle?

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Answer 1 · 2016-02-05T14:42:22

#"Area" = 20 " units"^2#

Let #alpha# be the angle opposite to the side #A#, #beta# be the angle opposite to the side #B# and #gamma# be the angle opposite to the side #C#.

Thus, you have:

#A = 10#, #B = 8#, #beta= (11 pi)/24# and #alpha = (3 pi)/8#.

Let's find out the length of the third angle first.

As the sum of all three angles in the triangle must be #180^@ = pi#, you know that

#gamma = pi - alpha - beta = pi - (11 pi)/24 - (3 pi)/8 = pi/6#

Now you have the sides #A# and #B# and #gamma#, the angle between those two sides. With this information, you can use the formula

#"Area" = 1/2 A * B * sin gamma#

#= 1/2 * 10 * 8 * sin(pi/6)#

#= 5 * 8 * 1/2#

#= 20 " units"^2#