using the half angle method solve the triangle for whick the given parts are a=27 b=21 and c=24

To use the half angle method to solve a triangle, we need to know the lengths of at least two sides and the measure of one angle. However, in this case, you have provided the lengths of all three sides of the triangle (a=27, b=21, and c=24).

Unfortunately, the half angle method is not applicable in this scenario. The half angle method is used to find the angles of a triangle when the lengths of the sides are given and the measures of the angles are unknown.

In this case, with all three side lengths given, we will need to use a different method to solve the triangle. One commonly-used method is the Law of Cosines.

The Law of Cosines states that for any triangle with sides of lengths a, b, and c opposite angles A, B, and C respectively, the following equation holds:

c^2 = a^2 + b^2 - 2ab * cos(C)

In your case, we can plug in the given side lengths into the equation:

24^2 = 27^2 + 21^2 - 2 * 27 * 21 * cos(C)

Simplifying this equation, we end up with:

576 = 729 + 441 - 1134 * cos(C)

Rearranging the terms, we get:

-594 = -1134 * cos(C)

Dividing both sides of the equation by -1134, we find:

cos(C) ≈ 0.5244

To find the measure of angle C, we can now use the inverse cosine function or the arccosine function.

C = arccos(0.5244)

Using a calculator or trigonometric tables, we find that C is approximately 58.89 degrees.

We can now find the remaining angles by using the Law of Sines or the Law of Cosines with the newly found angle C, and then use these angles to find any missing side lengths if needed.