Two corners of a triangle have angles of # (5 pi )/ 12 # and # pi / 4 #. If one side of the triangle has a length of 9, what is the longest possible perimeter of the triangle?

Question

Two corners of a triangle have angles of #  (5  pi  )/  12  # and #  pi  /  4  #. If one side of the triangle has a length of 9, what is the longest possible perimeter of the triangle?

Mason Adams · Accepted Answer

Longest possible perimeter #color(crimson)(P = 33.21#

#hat A = (5pi)/12, hat B = pi/4, hat C = pi/3#

Least angle #pi/4# should correspond to the side of length 9.

Applying Law of Sines,

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

#a = (b sin A) / sin B = (9 * sin ((5pi)/12)) / sin (pi/4) = 12.29#

#c = (9 sin(pi/3)) / sin(pi/4) = 12.02#

Longest possible perimeter #P = 9 + 12.29 + 12.02 = 33.21#

Wyatt Anderson · Answer

undefined To find the longest possible perimeter of the triangle, we need to consider the sum of the lengths of the other two sides.

Let's denote the angles of the triangle as $ A $, $ B $, and $ C $, where $ A $ is $ \frac{5\pi}{12} $ and $ B $ is $ \frac{\pi}{4} $.

Since the sum of the angles in a triangle is $ \pi $ radians, we can find the third angle $ C $ by subtracting the sum of the given angles from $ \pi $:

$$ C = \pi - \left(\frac{5\pi}{12} + \frac{\pi}{4}\right) $$

Now, we can use the Law of Sines to find the lengths of the other two sides of the triangle:

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

Given that one side, say side $ a $, has a length of 9, we can find the lengths of the other sides using the Law of Sines.

Once we find the lengths of the other two sides, we can calculate the perimeter of the triangle by adding all three side lengths together.

Finally, we select the longest possible perimeter among the calculated values.