In the given triangle ABC, angle A = 75 degrees, angle C = 30 degrees, and the side AC = 28. Find the length of side AB.

Solution:

• First, find the third angle of the triangle, angle B. The sum of angles in a triangle is 180 degrees. So, angle B = 180 - (angle A + angle C) = 180 - (75 + 30) = 180 - 105 = 75 degrees.
• Since angle A and angle B are equal (both 75 degrees), the triangle ABC is an isosceles triangle. In an isosceles triangle, the sides opposite to equal angles are equal.
• The side opposite to angle C is AB. The side opposite to angle B is AC.
• Since angle A = angle B, then the side opposite to angle A (which is BC) is equal to the side opposite to angle B (which is AC). Thus, BC = AC = 28.
• However, we need to find the length of side AB, which is opposite to angle C. Since angle A = angle B, the sides opposite to these angles are equal: BC = AC. This is incorrect based on the diagram.
• Let's re-evaluate. We have angle A = 75 degrees, angle C = 30 degrees, and angle B = 75 degrees. Since angle A = angle B, the sides opposite to these angles are equal. The side opposite to angle A is BC. The side opposite to angle B is AC. Therefore, BC = AC = 28. This is still not right.
• Let's check the angles again. Angle A = 75, Angle C = 30, Angle B = 180 - (75+30) = 75. So, Angle A = Angle B = 75 degrees.
• In a triangle, the sides opposite equal angles are equal. The side opposite angle A is BC. The side opposite angle B is AC. Therefore, BC = AC.
• The diagram shows AC = 28. So, BC = 28. We need to find AB.
• Let's use the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Here, a=BC, b=AC, c=AB.
• We have a = 28, b = 28, C = 30 degrees, A = 75 degrees, B = 75 degrees.
• We need to find c (AB). Using the Law of Sines: c/sin(C) = b/sin(B)
• AB / sin(30 degrees) = AC / sin(75 degrees)
• AB / 0.5 = 28 / sin(75 degrees)
• sin(75 degrees) = sin(45 + 30) = sin(45)cos(30) + cos(45)sin(30) = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2) = (sqrt(6) + sqrt(2))/4 ≈ (2.449 + 1.414)/4 = 3.863/4 ≈ 0.966
• AB = 0.5 * (28 / 0.966)
• AB ≈ 0.5 * 29.0
• AB ≈ 14.5

Final Answer: The length of side AB is approximately 14.5.