Question

With the usual notations, if the lengths of the sides of the triangle are 3 units, 5 units and 7 units, then the largest angle of the triangle is

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

Hint:

The largest angle is opposite the longest side; use the cosine rule.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The largest angle lies opposite the longest side, which is 7. Use the cosine rule.

Step 2: Key Formula or Approach
\[ \cos\theta=\frac{a^2+b^2-c^2}{2ab} \]
with \(a=3\), \(b=5\), \(c=7\).

Step 3: Calculation
\[ \cos\theta=\frac{9+25-49}{2\times3\times5}=\frac{-15}{30}=-\frac12 \]
\[ \theta=\frac{2\pi}{3} \]

Step 4: Check the options
\(\pi/2\) would need \(c^2=34\). \(\pi/3\) and \(\pi/4\) are acute, which cannot be the largest angle here because \(3^2+5^2<7^2\) shows an obtuse angle. The answer is (C).

Final Answer:
The largest angle has cosine -1/2, so it is \(\dfrac{2\pi}{3}\), option (C). \[ \boxed{\frac{2\pi}{3}} \]