Two corners of a triangle have angles of # (7 pi )/ 12 # and # pi / 8 #. If one side of the triangle has a length of # 1 #, what is the longest possible perimeter of the triangle?

Question

Two corners of a triangle have angles of #  (7  pi  )/  12  # and #  pi  /  8  #. If one side of the triangle has a length of # 1  #, what is the longest possible perimeter of the triangle?

Ella Armstrong · Accepted Answer

Longest possible perimeter of the triangle is #color(blue)(5.5972)#

#A = (7pi)/12, B = pi /8, C = pi - (7pi)/12 - pi / 8 = (7pi)/24#

Sideof length of 1 should correspond to smallest angle (pi/8) to get the longest perimeter.

#a / sin ((7pi)/12) = 1 / sin (pi/8) = c / sin ((7pi)/24)#

#a = (1 * sin ((7pi)/12)) / sin (pi/8) = 2.5241#

#c = (1 * sin ((7pi)/24)) / sin (pi/8) = 2.0731#

Longest possible perimeter of the triangle

#P = (a + b + c) / 2 = (2.5241 + 1 + 2.0731) / 2 = color (blue)(5.5972)#

Autumn Armstrong · Answer

undefined To find the longest possible perimeter of the triangle, we need to determine the length of the remaining sides and then calculate the sum of all three sides.

Let's denote the angles of the triangle as $ \theta_1 = \frac{7\pi}{12} $ and $ \theta_2 = \frac{\pi}{8} $. The third angle can be found by subtracting the sum of the other two angles from $ \pi $:

$$ \theta_3 = \pi - \left(\frac{7\pi}{12} + \frac{\pi}{8}\right) $$

Next, we can use the Law of Sines to find the lengths of the sides. Let $ a $, $ b $, and $ c $ be the lengths of the sides opposite angles $ \theta_1 $, $ \theta_2 $, and $ \theta_3 $ respectively. Then:

$$ \frac{a}{\sin\theta_1} = \frac{b}{\sin\theta_2} = \frac{c}{\sin\theta_3} $$

Given that one side has a length of 1, we can set $ b = 1 $ and solve for $ a $ and $ c $. After calculating $ a $ and $ c $, we can find the perimeter of the triangle:

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

Once we've computed the perimeter, we can check if it's the longest possible by considering different combinations of angles. Since there are only two angles given, we have limited options to consider.