Two corners of a triangle have angles of # (3 pi )/ 8 # and # ( pi ) / 2 #. If one side of the triangle has a length of # 7 #, what is the longest possible perimeter of the triangle?

Question

Two corners of a triangle have angles of #  (3  pi  )/  8  # and #  ( pi ) /  2  #. If one side of the triangle has a length of # 7  #, what is the longest possible perimeter of the triangle?

Aurora Ayala · Accepted Answer

Longest possible perimeter of the triangle is 42.1914

Given triangle is a right angle triangle as one of the angles is #pi/2# Three angles are #pi/2, (3pi)/8, pi/8# To get the longest perimeter, the side of length 7 should correspond to angle #pi8# (smallest angle). #:. a / sin A = b / sin B = c / sin C# #7/sin (pi/8) = b/sin ((3pi)/8) =c/sin (pi/2)# #b = (7 * sin ((3pi)/8)) / (sin (pi/8)) = 16.8995# #c =( 7 * sin (pi/2)) / sin (pi/8) = 18.2919# Longest possible perimeter #= (a + b + c) = 7 + 16.8995 + 18.2919 = 42.1914#

Ella Armstrong · Answer

undefined To find the longest possible perimeter of the triangle, we need to consider the triangle with its two given angles and one side length of 7.

The sum of the angles in a triangle is always $ \pi $ radians (or 180 degrees).

So, if we subtract the given angles from $ \pi $, we get the measure of the third angle.

$ \pi - \frac{3\pi}{8} - \frac{\pi}{2} = \pi - \frac{3\pi}{8} - \frac{4\pi}{8} = \pi - \frac{7\pi}{8} $

Now, we know the measure of all three angles of the triangle.

Using the Law of Sines, we can find the lengths of the other two sides of the triangle.

Then, we can calculate the perimeter of the triangle by adding all three side lengths together.

Finally, we can maximize the perimeter by maximizing the lengths of the other two sides of the triangle.

Thus, the longest possible perimeter of the triangle can be calculated accordingly.