In triangle ABC, side a = 7 cm, side b = 5 cm, and the included angle C = 60 degrees. Find the exact length of side c using the Law of Cosines.

Answer and explanation

Correct answer: D. sqrt(39) cm

  1. Step 1: The Law of Cosines states: c^2 = a^2 + b^2 - 2ab cos(C).
  2. Step 2: Substitute the given measurements: a = 7, b = 5, C = 60 degrees (cos 60 = 1/2).
  3. Step 3: c^2 = 7^2 + 5^2 - 2(7)(5) cos(60)
  4. Step 4: c^2 = 49 + 25 - 2(35)(1/2) = 74 - 35 = 39.
  5. Step 5: Take the square root: c = sqrt(39) cm.
Answer
sqrt(39) cm.
Why not the others: 'sqrt(74) cm' forgets to subtract the 2ab cos(C) term entirely; 'sqrt(109) cm' adds instead of subtracts (74 + 35 = 109); '6 cm' is an approximation sqrt(36).

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