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Answer:
a / sin(A) = b / sin(B) = c / sin(C)
Where a, b, and c are the lengths of the sides opposite angles A, B, and C, respectively.
Since we know the perimeter of the triangle is 300 meters, we can represent the lengths of the sides as follows:
a + b + c = 300
We can now use the Law of Sines to find the length of side c, which is opposite the largest angle C:
c / sin(C) = a / sin(A)
We know that sin(A) = sin(36 deg) and sin(C) = sin(86 deg). We can now solve for c:
c = (a * sin(C)) / sin(A)
Since we don't have the individual lengths of sides a and b, we can't directly calculate the length of side c. However, we can use the fact that the sum of the angles in a triangle is 180 degrees to find the measure of angle A:
A + B + C = 180 deg
36 deg + 58 deg + 86 deg = 180 deg
This confirms that our given angles are correct. Now, we can use the fact that the perimeter is 300 meters to find the length of side c:
a + b + c = 300
a + b = 300 - c
We can now substitute this expression into the Law of Sines equation:
c / sin(C) = (300 - c) / sin(A)
Solving for c, we get:
c = (300 * sin(C)) / (sin(A) + sin(B) + sin(C))
Now, we can plug in the values for sin(A), sin(B), and sin(C) to find the length of side c:
c = (300 * sin(86 deg)) / (sin(36 deg) + sin(58 deg) + sin(86 deg))
Using a calculator, we find:
sin(36 deg) ≈ 0.5878
sin(58 deg) ≈ 0.8480
sin(86 deg) ≈ 0.9996
c ≈ (300 * 0.9996) / (0.5878 + 0.8480 + 0.9996)
c ≈ 299.88 / 2.4354
c ≈ 123.16 meters
Therefore, the side opposite the biggest angle, angle C, is approximately 123.16 meters long.