The side of an isosceles triangle is 12.8 cm.Calculate the area of the triangle if you know that the angle at the base is …

Let’s designate this triangle ABC, AC – base, AB = BC – lateral sides.
To solve the problem, we will use the formula for the area of ​​a triangle through two sides and the sine of the angle between them.
S ABC = 1/2 * AB * BC * sin B.
For each specific case, we find the angle B at the apex of an isosceles triangle.
1.
∠ B = 180 ° – 2 * 30 ° = 120 °.
S ABC = 1/2 * AB * BC * sin 120 ° = 1/2 * 12.8 * 12.8 * √3 / 2 = 70.944801 ≈ 71 (cm²).
2.
∠ B = 180 ° – 2 * 45 ° = 90 °.
S ABC = 1/2 * AB * BC * sin 90 ° = 1/2 * 12.8 * 12.8 * 1 = 81.92 (cm²).
3.
∠ B = 180 ° – 2 * 60 ° = 60 °.
S ABC = 1/2 * AB * BC * sin 60 ° = 1/2 * 12.8 * 12.8 * √3 / 2 = 70.944801 ≈ 71 (cm²).
Answer: in each of the three cases, the area is 71 cm², 81.92 cm², 71 cm²