Question

If the sides of a triangle are p, q and \[\sqrt {{p^2} + pq + {q^2}} \], then the biggest angle is
A. \[\pi /2\]
B. \[2\pi /3\]
C. \[5\pi /4\]
D. \[7\pi /4\]
 E. \[5\pi /3\]

Answer 1

Hint: To get the biggest angle in the given triangle, we shall apply the triangle inequality theorem. The greatest sides will be found, and the angle opposite the biggest side will be the maximum angle of that triangle.

FORMULA USED:
\[\cos = \dfrac{{{a^2} + {c^2} - {b^2}}}{{2ac}}\]

Complete step by step solution: Given that the triangle has three sides p,q and \[\sqrt {{p^2} + pq + {q^2}} \].
The largest side of the triangle is considered as \[\sqrt {{p^2} + pq + {q^2}} \]
Let the largest side of the triangle’s angle be \[\theta \]
Then, the equation becomes,
\[\cos \theta = \dfrac{{{p^2} + {q^2} - {p^2} - pq - {q^2}}}{{2pq}}\]
Then, which is equal to
\[ - \dfrac{1}{2} = \cos (\dfrac{{2\pi }}{3})\]
Hence, the angle of the triangle is,
\[\theta = \dfrac{{2\pi }}{3}\].

So, Option ‘A’ is correct

Note: The "largest" angle in a triangle is the angle created by the triangle's sides, and it may be determined using the formula.
Each of the smaller angles can be added up to find a larger angle. The largest angle in this equation would be \[180^\circ - 90^\circ \], or \[135.5\] degrees, because \[{p^2} + pq + {q^2} = 180^\circ .\]