# Two corners of a triangle have angles of # (5 pi )/ 8 # and # ( pi ) / 6 #. If one side of the triangle has a length of # 12 #, what is the longest possible perimeter of the triangle?

Longest possible perimeter of the triangle

Applying the law of Sines,

Longest possible perimeter of the triangle

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To find the longest possible perimeter of the triangle, we need to maximize the sum of the lengths of the other two sides. Using the Law of Sines, we can find the lengths of these sides.

First, let's find the third angle of the triangle:

( \text{Third angle} = \pi - \left(\frac{5\pi}{8} + \frac{\pi}{6}\right) = \frac{9\pi}{24} = \frac{3\pi}{8})

Now, using the Law of Sines:

( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C})

Where (a), (b), and (c) are the lengths of the sides opposite angles (A), (B), and (C) respectively.

Given that side (c = 12), and angle (C = \frac{3\pi}{8}), we can use this to find the lengths of the other two sides:

(b = \frac{c \cdot \sin B}{\sin C} = \frac{12 \cdot \sin\left(\frac{5\pi}{8}\right)}{\sin\left(\frac{3\pi}{8}\right)})

(a = \frac{c \cdot \sin A}{\sin C} = \frac{12 \cdot \sin\left(\frac{\pi}{6}\right)}{\sin\left(\frac{3\pi}{8}\right)})

Now, calculate (b) and (a), then find the perimeter by summing up all the sides. Finally, choose the maximum perimeter among the values of (b) and (a).

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