Final answer:
To find the perimeter of triangle ABC, use the Law of Sines with the provided angles to calculate the lengths of sides BC and CA, and then sum the lengths of all three sides.
Step-by-step explanation:
The student asked to find the perimeter of triangle ABC, with given side AB = 12, angle A = 60 degrees, and angle C = 45 degrees. To find the perimeter, we must first recognize that the sum of the angles in a triangle is 180 degrees. Therefore, angle B = 180 - 60 - 45 = 75 degrees. Using the Law of Sines, we can find the lengths of the other sides (BC and CA).
Let side AC be c and side BC be a. The Law of Sines states:
- sin(A)/AB = sin(B)/a = sin(C)/c
So we have:
- sin(60)/12 = sin(75)/a = sin(45)/c
We can calculate a and c using:
- a = (sin(75) * 12) / sin(60)
- c = (sin(45) * 12) / sin(60)
Once we have the lengths of a and c, we can simply add the lengths of the three sides to find the perimeter of the triangle:
Perimeter = AB + BC + CA = 12 + a + c