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: Approach
Check the type of triangle first with Pythagoras, then find the angle.

Step 2: Type
$3^2+5^2=34$ and $7^2=49$. Since $34<49$ the largest angle is obtuse.

Step 3: Angle
Let the largest angle be $C$:
\[ 49=9+25-30\cos C\Rightarrow\cos C=-\frac{15}{30}=-\frac12 \]
So $C=120^\circ=\dfrac{2\pi}{3}$. This is option (C).

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