Two corners of a triangle have angles of #(7 pi ) / 12 # and # (3 pi ) / 8 #. If one side of the triangle has a length of #6 #, what is the longest possible perimeter of the triangle?

Question

Two corners of a triangle have angles of #(7 pi ) /  12  # and # (3 pi  ) /  8  #.  If one side of the triangle has a length of #6  #, what is the longest possible perimeter of the triangle?

Savannah Aguilar · Accepted Answer

Longest possible perimeter P = 92.8622

Given #: /_ C = (7pi) /12, /_B = (3pi)/8# # /_A = (pi - (7pi) /12 - (3pi)/8 ) = pi / 24# To get the longest perimeter, we should consider the side corresponding to the angle that is the smallest. #a / sin A = b / sin B = c / sin C# #6 / sin (pi/24) = b / sin ((3pi)/8) = c / sin ((7pi)/12)# #:. b = (6 * sin ((3pi)/8)) / sin (pi/24) = 42.4687# #c = (6 * sin ((7pi)/12))/sin (pi/24) = 44.4015# Longest possible perimeter #P = 6 + 42.4687 + 44.4015 = 92.8622#

Savannah Aguilar · Answer

undefined To find the longest possible perimeter of the triangle, we can use the law of sines.

The law of sines states: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$, where $a$, $b$, and $c$ are the lengths of the sides opposite angles $A$, $B$, and $C$, respectively.

Given that one side has a length of $6$ and the angles are $\frac{7\pi}{12}$ and $\frac{3\pi}{8}$, we can find the lengths of the other two sides using the law of sines.

Let's denote the unknown sides as $b$ and $c$, and the corresponding angles as $B$ and $C$, respectively.

Then, we have:

$\frac{6}{\sin(\frac{7\pi}{12})} = \frac{b}{\sin(\frac{3\pi}{8})}$

Solving for $b$, we get: $b = 6 \times \frac{\sin(\frac{3\pi}{8})}{\sin(\frac{7\pi}{12})}$

Similarly, we can find $c$ using the law of sines:

$\frac{6}{\sin(\frac{7\pi}{12})} = \frac{c}{\sin(\pi - \frac{7\pi}{12} - \frac{3\pi}{8})}$

Solving for $c$, we get: $c = 6 \times \frac{\sin(\pi - \frac{7\pi}{12} - \frac{3\pi}{8})}{\sin(\frac{7\pi}{12})}$

Once we have the lengths of all three sides, we can calculate the perimeter of the triangle and determine the longest possible perimeter.