A triangle has two corners with angles of # pi / 2 # and # ( pi )/ 8 #. If one side of the triangle has a length of #8 #, what is the largest possible area of the triangle?

Question

A triangle has two corners with angles of #  pi  /  2  # and # ( pi  )/  8   #.  If one side of the triangle has a length of #8  #, what is the largest possible area of the triangle?

Wyatt Anderson · Accepted Answer

Largest possible area of the triangle is 77.2548

Given are the two angles #(pi)/2# and #pi/8# and the length 1 The remaining angle: #= pi - ((pi)/2) + pi/8) = (3pi)/8# I am assuming that length AB (1) is opposite the smallest angle. Using the ASA Area#=(c^2*sin(A)*sin(B))/(2*sin(C)# Area#=8^2*sin(pi/2)*sin((3pi)/8))/(2*sin(pi/8))# Area#=77.2548#

Parker Allen · Answer

undefined To find the largest possible area of the triangle, given that it has two angles of π/2 and π/8 and one side length of 8 units, you can use trigonometry and the area formula for a triangle.

First, let's denote the two known angles as A and B. Angle A is π/2 radians, and angle B is π/8 radians.

Since the sum of angles in a triangle is π radians, you can find the third angle C by subtracting the sum of angles A and B from π:

$$ C = π - (A + B) $$

$$ C = π - (\frac{π}{2} + \frac{π}{8}) $$

Now, find the value of angle C.

Once you have all three angles, you can use the Law of Sines to find the lengths of the other two sides. Then, you can use the formula for the area of a triangle, which is:

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$

After finding the lengths of the other two sides, you can calculate the area of the triangle using the formula mentioned above. The largest possible area corresponds to the triangle with the greatest height.