A triangle has two corners with angles of # pi / 4 # and # (5 pi )/ 8 #. If one side of the triangle has a length of #9 #, what is the largest possible area of the triangle?

Question

A triangle has two corners with angles of #  pi  /  4  # and # (5 pi  )/  8   #.  If one side of the triangle has a length of #9  #, what is the largest possible area of the triangle?

Wyatt Anderson · Accepted Answer

Largest possible area of the triangle is 69.1378

Given are the two angles #(pi)/4# and #(5pi)/8# and the length 9 The remaining angle: #= pi - ((5pi)/8) + pi/4) = (pi)/8# I am assuming that length AB (9) is opposite the smallest angle. Using the ASA Area#=(c^2*sin(A)*sin(B))/(2*sin(C)# Area#=( 9^2*sin(pi/4)*sin((5pi)/8))/(2*sin(pi/8))# Area#=69.1378#

Aurora Ayala · Answer

undefined To find the largest possible area of the triangle, we can use the formula for the area of a triangle given two sides and the angle between them:

Area = 1/2 * a * b * sin(C)

where a and b are the lengths of the two sides, and C is the angle between them.

Let's denote the angle with π/4 radians as A and the angle with (5π)/8 radians as B. We know that the sum of the angles in a triangle is π radians, so the third angle C can be calculated as:

C = π - A - B

Now, we can calculate the area of the triangle for different configurations where one side is 9 units long and the other two sides are determined by the angles A and B. We can then find the maximum area among these configurations.

Let's start by calculating the area for the given angles A = π/4 and B = (5π)/8:

C = π - π/4 - (5π)/8 = π/8

Now, using the formula for the area of a triangle:

Area = 1/2 * 9 * 9 * sin(π/8)

Calculate the sin(π/8) and then multiply it by 1/2 * 9 * 9 to find the maximum area of the triangle.