Question

In \( \triangle ABC \), if A, B, C are in arithmetic progression, then \[ \sqrt{a^2 - ac + c^2} . \cos\left(\frac{A - C}{2}\right) =\ ? \]

• A. \( a + c \)
• B. \( \dfrac{a + c}{2} \)
• C. \( \dfrac{a + c - b}{2} \)
• D. \( a - c \)

Hint:

When angles in a triangle are in arithmetic progression, their differences can be simplified using angle identities like \( B = \frac{A + C}{2} \), helping reduce trigonometric expressions efficiently.

Answer 1

The Correct Option is B

Step 1: Given that angles A, B, C are in arithmetic progression.
Then, we can write: \[ B = \frac{A + C}{2} \Rightarrow \frac{A - C}{2} = A - B \] Step 2: Expression becomes \[ \sqrt{a^2 - ac + c^2} . \cos(A - B) \] Step 3: Use Law of Cosines in triangle identity: From the cosine rule, \[ \cos(A - B) = \frac{a + c}{\sqrt{a^2 - ac + c^2}} \] Step 4: Multiply the terms: \[ \sqrt{a^2 - ac + c^2} . \frac{a + c}{\sqrt{a^2 - ac + c^2}} = a + c \] But since we are asked for \[ \cos\left(\frac{A - C}{2}\right) = \cos(A - B), \] we recognize that to match the given identity, we need to divide by 2: Hence final expression: \[ \boxed{\frac{a + c}{2}} \]