Two corners of a triangle have angles of # (5 pi )/ 8 # and # ( pi ) / 2 #. 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 #  (5  pi  )/  8  # and #  ( pi ) /  2  #. If one side of the triangle has a length of # 6  #, what is the longest possible perimeter of the triangle?

Avery Allen · Accepted Answer

Perimeter #= a + b + c = color (green)(36.1631)# Sum of the three angles of a triangle is equal to #180^0 or pi# As the sum of the given two angles is #=(9pi)/8# which is greater than #pi#, the given sum needs correction. It’s assumed that the two angles be #color(red)((3pi)/8 & pi/2)# #/_A = (5pi)/8, /_B = pi/2,# # /_C = pi -( ((3pi)/8 )- (pi/2)) =pi - (7pi)/8 = pi/8# To get the longest perimeter, length 6 should correspond to the smallest #/_C = pi/8# #a / sin( /_A )= b / sin( /_B )= c / sin( /_C)# #a / sin ((3pi)/8) = b / sin (pi/2) = 6 / sin (pi/8)# #a = (6 * sin ((3pi)/8)) / sin (pi /8)# #a = (6 * 0.9239)/ 0.3827 =color(blue)( 14.485)# #b = (6 * sin (pi/2))/ sin (pi/8)# #b = 6 / 0.3827 = color(blue)(15.6781)# Perimeter #= a + b + c = 6 + 14.485 + 15.6781 = color (green)(36.1631)#

Aurora Ayala · Answer

undefined To find the perimeter of the triangle, you need to determine the lengths of the other two sides using trigonometric ratios. Since you have two angles of the triangle, you can use trigonometric functions to find the lengths of the sides opposite to those angles.

Let's denote the given angles as $ \frac{5\pi}{8} $ and $ \frac{\pi}{2} $. The sum of the angles in a triangle is $ \pi $ radians.

Using the fact that the sum of angles in a triangle is $ \pi $ radians, you can find the measure of the third angle:

$$ \text{Third angle} = \pi - \left(\frac{5\pi}{8} + \frac{\pi}{2}\right) $$

With the third angle known, you can use trigonometric ratios to find the lengths of the other two sides of the triangle. For example, you can use the sine and cosine functions:

$$ \text{sin}(\text{angle}) = \frac{\text{opposite side}}{\text{hypotenuse}} $$
$$ \text{cos}(\text{angle}) = \frac{\text{adjacent side}}{\text{hypotenuse}} $$

Given the length of one side ($6$), you can use trigonometric ratios to find the lengths of the other two sides. Once you have all three side lengths, you can calculate the perimeter by adding them together.

After finding the lengths of all three sides, add them together to get the perimeter of the triangle. The longest possible perimeter occurs when the third angle is the smallest, resulting in the longest possible side lengths opposite the given angles.