Question

In \(\triangle ABC\), \((a + b) \cdot \cos C + (b + c) \cdot \cos A + (c + a) \cdot \cos B\) is equal to .........

• A. \( a - b + c \)
• B. \( a + b - c \)
• C. \( a + b + c \)
• D. \( a - b - c \)

Hint:

In geometric expressions involving angles and sides, look for simplifications based on vector identities or trigonometric properties.

Answer 1

The Correct Option is C

Step 1: Recognize the formula.
The expression involves the sum of products of sides and cosines of angles. From vector algebra, this is equivalent to the dot product formula. The formula simplifies to the sum of sides \(a\), \(b\), and \(c\).

Step 2: Apply known properties.
Using the fact that in any triangle, the sum of side-related cosines produces a simplified result as \( a + b + c \).

Step 3: Conclude.
The correct expression is \(a + b + c\), so option (iii) is correct.

Final Answer: \[ \boxed{a + b + c} \]