In any , the value of 2ac ( A-B + C 2 ) is: - Vidyadip

MCQ

In any $\triangle\text{ABC},$ the value of $2\text{ac}\sin\Big(\frac{\text{A}-\text{B + C}}{2}\Big)$ is:

A
$\text{a}^2+\text{b}^2-\text{c}^2$
✓
$\text{c}^2+\text{a}^2-\text{b}^2$
C
$\text{b}^2-\text{c}^2-\text{a}^2$
D
$\text{c}^2-\text{a}^2-\text{b}^2$

✓

Answer

Correct option: B.

$\text{c}^2+\text{a}^2-\text{b}^2$

In $\triangle\text{ABC},$
$\text{A + B + C}=\pi$ (Angle sum property)
$\Rightarrow\text{A + C}=\pi-\text{B}$
$\therefore2\text{ac}\sin\Big(\frac{\text{A}-\text{B + C}}{2}\Big)$
$=2\text{ac}\sin\Big(\frac{\pi-2\text{B}}{2}\Big)$
$=2\text{ac}\sin\Big(\frac{\pi}{2}-\text{B}\Big)$
$=2\text{ac}\cos\text{B}$
$=2\text{ac}\Big(\frac{\text{c}^2+\text{a}^2-\text{b}^2}{2\text{ca}}\Big)$ (Using cosine rule)
$=\text{c}^2+\text{a}^2-\text{b}^2$
Hence, the correct answer is option (b).

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