Two corners of a triangle have angles of #(3 pi ) / 8 # and # pi / 4 #. If one side of the triangle has a length of #4 #, what is the longest possible perimeter of the triangle?

Question

Two corners of a triangle have angles of #(3 pi ) /  8  # and #  pi   /  4  #.  If one side of the triangle has a length of #4  #, what is the longest possible perimeter of the triangle?

James Atkins · Accepted Answer

Longest possible perimeter of the triangle is 14.4526

Given are the two angles #(pi)/4# and #(3pi)/8# and the length 1 The remaining angle: #= pi - (((pi)/4) + (3pi)/8) = (3pi)/8# I am assuming that length AB (4) is opposite the smallest angle #a / sin A = b / sin B = c / sin C# #4 / sin ((pi)/4) = b / sin ((3pi) /8) = c / ((3pi) / 8)# #b = (4*sin((3pi)/8)) / sin ((pi) /4) = 5.2263# #c = (4*sin ((3pi)/8)) / sin ((pi)/4) = 5.2263# Longest possible perimeter of the triangle is =# (a+b+c) = (4+5.2263+5.2263) = 14.4526#

Noah Atkinson · Answer

undefined To find the longest possible perimeter of the triangle, you can use the law of sines to determine the lengths of the other sides of the triangle. Then, calculate the perimeter by adding the lengths of all three sides.

Given the angles $\frac{3\pi}{8}$ and $\frac{\pi}{4}$, you can find the third angle by subtracting the sum of the given angles from $\pi$ (since the sum of angles in a triangle is $\pi$). Then, use the law of sines to find the lengths of the other sides.

Let $a$, $b$, and $c$ represent the lengths of the sides of the triangle.

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

Given that one side has a length of 4, you can find the lengths of the other sides using the law of sines and the given angles. Then, calculate the perimeter by adding the lengths of all three sides.

$$ \text{Perimeter} = a + b + c $$

Find $C$, the third angle:

$$ C = \pi - \left(\frac{3\pi}{8} + \frac{\pi}{4}\right) $$

Use the law of sines to find the lengths of the other sides:

$$ \frac{\sin\left(\frac{3\pi}{8}\right)}{4} = \frac{\sin\left(\frac{\pi}{4}\right)}{b} = \frac{\sin C}{c} $$

Calculate $b$ and $c$ using the law of sines.

Once you have $a$, $b$, and $c$, calculate the perimeter of the triangle by adding the lengths of all three sides. This will give you the longest possible perimeter of the triangle.