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# Find the area of an oblique triangle

Question:

In triangle ABC, AB = 9, AC = 8 and BC = 6. What is the largest angle in triangle ABC? What is the area of the triangle ABC?

Solution:

The largest angle opposite the longest side. Because the longest side is AB, so, the largest angle is the angle C. Now, we use the cosine law to find the angle C.

cos C = (a2 + b2 - c2)/2ab

In which, a is the side opposite the angle A. Because the angle A is opposite the side BC and BC = 6, so, a = 6. b is the side opposite the angle B. Because the angle B is opposite the side AC and AC = 8, so, b = 8. C is the side opposite the angle C. Because the angle C is opposite the side AB and AB = 9, so c = 9.

cos C = (6 × 6 + 8 × 8 - 9 × 9)/2 × 6 × 8 = (36 + 64 - 81)/2 × 6 × 8 = 19/96 = 0.1979
angle C = arc cos 0.1979 = 78.6o
the formula of the area of a triangle is: A = (ab/2) sin C
A = (1/2) ab sin C = (1/2) × 6 × 8 × sin 78.6o = (1/2) × 6 × 8 × 0.98 = 23.52

Therefore, the largest angle is the angle C and the angle C is 78.6o. The area of the triangle is 23.52. Looking for more detail? Please watch the video.