Two corners of a triangle have angles of # (5 pi )/ 12 # and # ( pi ) / 3 #. 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 #  (5  pi  )/  12  # and #  ( pi ) /  3  #. If one side of the triangle has a length of # 1  #, what is the longest possible perimeter of the triangle?

Jayden Jackson · Accepted Answer

Longest possible perimeter #color(orange)(P = 1 + 1.22 + 1.37 = 3.59# #hat A = (5pi)/12, hat B = pi/3, hat C = pi/4# Side 1 should correspond to #hat C = pi/4# the least angle to get the longest perimeter. As per Law of Sines, #a / sin A = b / sin B = c / sin C# #:. a = (sin ((5pi)/12) * 1) / sin (pi/4) = 1.37# #b = (sin (pi/3) * 1) / sin(pi/4) = 1.22# Longest possible perimeter #color(orange)(P = 1 + 1.22 + 1.37 = 3.59#

Violet Aldrich · Answer

undefined To find the longest possible perimeter of the triangle, we need to consider the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

First, let's identify the third angle of the triangle. The sum of angles in a triangle is always π radians (180 degrees). So, the third angle can be found by subtracting the given angles from π radians:

Third angle = π - (5π/12 + π/3)

Once we have the third angle, we can use the Law of Sines to find the lengths of the other two sides of the triangle. Then, we can calculate the perimeter by adding the lengths of all three sides.

Let's calculate:

Third angle = π - (5π/12 + π/3)
             = π - (15π/36 + 12π/36)
             = π - (27π/36)
             = (36π - 27π)/36
             = 9π/36
             = π/4

Now, we use the Law of Sines to find the lengths of the other two sides:

sin(A)/a = sin(B)/b = sin(C)/c

Let's denote the side opposite the angle (5π/12) as 'a' and the side opposite the angle (π/3) as 'b'. We already know the length of side 'b' is 1 unit.

sin(5π/12)/a = sin(π/4)/1

We can solve for 'a':

sin(5π/12) = a * sin(π/4)
a = sin(5π/12) / sin(π/4)

We can use a calculator to find the values of sin(5π/12) and sin(π/4) and calculate 'a'.

Similarly, we can use the Law of Sines to find the length of side 'c' (opposite the angle π/3) and then calculate the perimeter by adding the lengths of all three sides.

Perimeter = a + b + c

Once we have the lengths of sides 'a', 'b', and 'c', we can substitute them into the formula for perimeter to find the longest possible perimeter of the triangle.