Final answer:
The measure of angle B in triangle ABC using the Law of Cosines and the given sides' lengths is approximately 58.2 degrees. The provided answer choices do not match this result, suggesting there may be a typo in the question or answer choices.
Step-by-step explanation:
To find the measure of angle B in triangle ABC with sides a=7, b=8, and c=9, you can use the Law of Cosines. This mathematical rule relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula for the Law of Cosines is:
c^2 = a^2 + b^2 - 2ab\(\cos(C)\), where C is the angle opposite side c.
By rearranging the formula to solve for \(\cos(C)\), we have:
\(\cos(C)\) = \(\frac{a^2 + b^2 - c^2}{2ab}\)
Substitute the given values:
\(\cos(B)\) = \(\frac{7^2 + 9^2 - 8^2}{2 \times 7 \times 9}\) = \(\frac{49 + 81 - 64}{126}\) = \(\frac{66}{126}\)
Calculate \(\cos(B)\):
\(\cos(B)\) = \(\frac{66}{126}\) = 0.52380952381
Find the angle B by taking the inverse cosine:
B = \(\cos^{-1}(0.52380952381)\)
This computation gives:
B ≈ 58.2 degrees
However, this result does not match any of the options provided. We need to check our calculations or assume there might be a typo in the question or the answer choices. The answer choices provided in the question suggest that option (b), 53.1 degrees, might be the intended correct answer, but based on the calculations above, this is not the case for the values given.