The correct answer is option is ii. a = 10√2, b = 5, c = 15, d = 5.
Case i:
1. Angle B: We know the value of angle C (60°) and that the angles in a triangle add up to 180°. Therefore, angle B = 180° - angle C - angle A = 180° - 60° - 30° = 90°.
2. Side b: Since angle B is 90°, triangle ABC is a right triangle. We also know the value of side c (15) and side a (10√3). We can use the Pythagorean theorem to find side b: b^2 = c^2 - a^2 = 15^2 - (10√3)^2 = 225 - 300 = -75. However, side lengths cannot be negative. Therefore, there is no solution for this case.
Case ii:
1. Angle B: Similar to case i, angle B = 180° - angle C - angle A = 180° - 60° - 30° = 90°.
2. Side b: Using the Pythagorean theorem again, we get b^2 = c^2 - a^2 = 15^2 - (10√2)^2 = 225 - 200 = 25. Therefore, side b = 5.
Case iii:
1. Angle B: Once again, angle B = 180° - angle C - angle A = 180° - 60° - 30° = 90°.
2. Side b: Using the Pythagorean theorem, we get b^2 = c^2 - a^2 = 15^2 - (10√3)^2 = 225 - 300 = -75. As in case i, there is no solution for this case because side lengths cannot be negative.
In summary:
- In case ii, the values of the variables are: a = 10√2, b = 5, c = 15, and d = 5.
- In cases i and iii, there is no solution because the Pythagorean theorem results in negative side lengths, which are not possible.