Two corners of a triangle have angles of # ( pi )/ 3 # and # ( pi ) / 4 #. If one side of the triangle has a length of # 9 #, what is the longest possible perimeter of the triangle?
Perimeter is
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To find the longest possible perimeter of the triangle given two angles of π/3 and π/4, and one side of length 9, follow these steps:
 Use the fact that the sum of angles in a triangle is π radians (180 degrees) to find the third angle.
 Once you have all three angles, use the Law of Sines to find the lengths of the other two sides.
 Since you want to maximize the perimeter, choose the largest possible values for the other two sides.
 Calculate the perimeter using the lengths of all three sides.
Let's proceed with the calculations:

The third angle of the triangle can be found by subtracting the sum of the given angles from π radians: ( \text{Third angle} = \pi  \left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \pi  \frac{7\pi}{12} = \frac{5\pi}{12} ).

Use the Law of Sines to find the lengths of the other two sides: ( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ), where ( a, b, c ) are the side lengths and ( A, B, C ) are the opposite angles.
Let's denote the side opposite the angle π/3 as ( a = 9 ), and let ( b ) and ( c ) be the other two sides.
For the angle π/3: ( \frac{9}{\sin(\pi/3)} = \frac{b}{\sin(5\pi/12)} ). Solve for ( b ): ( b = 9 \cdot \frac{\sin(5\pi/12)}{\sin(\pi/3)} ).

Repeat the same process for the angle π/4 to find ( c ): ( \frac{9}{\sin(\pi/4)} = \frac{c}{\sin(5\pi/12)} ). Solve for ( c ): ( c = 9 \cdot \frac{\sin(5\pi/12)}{\sin(\pi/4)} ).

Once you have the lengths of all three sides, calculate the perimeter: ( \text{Perimeter} = 9 + b + c ).
Calculate ( b ) and ( c ) using the given trigonometric ratios, then find the perimeter using the formula provided. This will give you the longest possible perimeter of the triangle.
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