Final answer:
To find the longest side of the triangle, utilize the Law of Sines with the given angles and the length of the shortest side. By setting a ratio between the shortest side and its opposite angle to the longest side and its opposite angle, and then solving for the unknown, we determine that the longest side is approximately 7.94 inches.
Step-by-step explanation:
To solve for the approximate length of the longest side of a triangle with angles measuring 35°, 63°, and 82°, and with the shortest side being 5 inches, we will use the Law of Sines. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in a triangle. This can be written as a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the side lengths, and A, B, C are the opposite angles.
First, we identify the longest side is opposite the largest angle, which is 82°. Let's denote the shortest side (5 inches) as side 'a' and angle 35° as angle A. Since side 'a' is opposite angle A, we can set up a ratio comparing it to the unknown longest side 'c' opposite the largest angle C (82°).
Thus, the Law of Sines formula in this context will be: 5/sin(35°) = c/sin(82°). After calculating the sines of both angles, we cross-multiply and solve for 'c':
c = 5 × sin(82°) / sin(35°)
By performing the calculations, we find the length of 'c' to be approximately 7.94 inches, which is the length of the longest side of the triangle.