How can we find the value of side B in triangle ABC where A=9, C=5, and angle B=120 degrees?

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In triangle ABC, we have the lengths of sides A and C, and the measure of angle B. To find the length of side B, we can employ the Law of Cosines, which is particularly useful in non-right triangles. The Law of Cosines states:

B2 = A2 + C2 – 2AC * cos(B)

where A, B, and C are the lengths of the sides opposite to angles A, B, and C respectively.

Given that:

  • A = 9
  • C = 5
  • B = 120 degrees

Now substituting the given values into the formula:

B2 = 92 + 52 – 2 * 9 * 5 * cos(1200)

Calculating each term:

  • 92 = 81
  • 52 = 25
  • cos(1200) = -0.5 (since 120 degrees is in the second quadrant)

Now, substituting these values back into the equation:

B2 = 81 + 25 – 2 * 9 * 5 * (-0.5)

This simplifies to:

B2 = 81 + 25 + 45

B2 = 151

Finally, to find B, we take the square root:

B = √151

This gives us the length of side B:

B ≈ 12.25

In conclusion, using the Law of Cosines is a systematic way to find the missing side in a triangle when you know the other sides and the included angle.

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