Final answer:
The given conditions of side lengths 3 inches and 8 inches with an included angle of 75° determine a unique triangle, as the configurations do not suggest the possibility of no triangle or multiple distinct triangles.
Step-by-step explanation:
The conditions given are side-side-angle (SSA), which typically do not determine a unique triangle. In some cases, depending on the lengths of the sides and the measure of the angle, SSA can result in no triangle or two distinct triangles. In the scenario with sides measuring 3 inches and 8 inches and an included angle of 75°, we can determine if a triangle is possible by using the Law of Sines to find the opposite side to the given angle.
If a / sin(A) = b / sin(B), where A is the given angle, a is the side opposite A, b is one of the given sides, and B is the angle opposite b, we can solve for B. However, since the given side lengths of 3 and 8 with an included angle of 75° do not satisfy this condition, the lack of ambiguity tells us that only one triangle can be formed with these given measurements.
The given conditions determine one unique triangle.
When given the lengths of two sides and the included angle of a triangle, we can use the Law of Cosines to determine the length of the third side. The Law of Cosines states that c^2 = a^2 + b^2 - 2ab cos(C), where c is the length of the third side and C is the included angle.
In this case, the length of one side is 3 in, the length of another side is 8 in, and the included angle is 75°. Plugging these values into the equation, we have c^2 = 3^2 + 8^2 - 2(3)(8)cos(75°). Solving this equation will give us the length of the third side.