
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
163.5k+ views
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 \[{p^2} + pq + {q^2} = 180^\circ .\]
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 \[{p^2} + pq + {q^2} = 180^\circ .\]
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Degree of Dissociation and Its Formula With Solved Example for JEE

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

Instantaneous Velocity - Formula based Examples for JEE

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations

NCERT Solutions for Class 11 Maths Chapter 8 Sequences and Series
