Question

In triangle ABC, with usual notations, if \((a+b+c)(a+b-c) = ab\), then the measure of angle C is...

• A. \(\frac{π}{2}\)
• B. \(\frac{2π}{3}\)
• C. \(\frac{5π}{6}\)
• D. \(\frac{3π}{4}\)

Hint:

Expand the product and use the cosine rule.

Answer 1

The Correct Option is B

Step 1: Rewrite:
$(a + b)^2 - c^2 = ab$ means $c^2 = a^2 + ab + b^2$.

Step 2: Compare with cosine rule:
$c^2 = a^2 + b^2 - 2ab\cos C$, so $-2\cos C = 1$, giving $\cos C = -\frac12$ and $C = 120^{\circ} = \frac{2\pi}{3}$.

Final Answer:
The angle is $\frac{2\pi}{3}$, option (B). \[ \boxed{\frac{2\pi}{3}} \]