If the sides of the triangle are p,q,√p2+q2+pq, then the greatest angle is:

Question

If the sides of the triangle are p,q,√p2+q2+pq, then the greatest angle is: (A)  (π /2)  (B)  (5π /4)  (C)  (2π /3)  (D)  (7π /4)

Tardigrade Admin · Accepted Answer

The longest side of triangle is √p2+q2+pq . Greatest angle will be opposite to longest side. Let θ be greatest angle, then cos θ =(p2+q2-p2-q2-pq/2pq)=-(1/2) ⇒ cos θ = cos (2π /3) ⇒ θ =(2π /3)

Q. If the sides of the triangle are $p, q, p^{2} + q^{2} + pq,$ then the greatest angle is:

$\frac{π}{2}$

$\frac{5 π}{4}$

$\frac{2 π}{3}$

$\frac{7 π}{4}$

$\frac{5 π}{3}$

The longest side of triangle is $p^{2} + q^{2} + pq$ .
Greatest angle will be opposite to longest side. Let $θ$ be greatest angle, then $cos θ = \frac{p ^{2} + q ^{2} - p ^{2} - q ^{2} - pq}{2 pq} = - \frac{1}{2}$
$\Rightarrow$ $cos θ = cos \frac{2 π}{3}$
$\Rightarrow$ $θ = \frac{2 π}{3}$

Q. If the sides of the triangle are p,q,p2+q2+pq​, then the greatest angle is:

2π​

45π​

32π​

47π​

35π​

The longest side of triangle is p2+q2+pq​ . Greatest angle will be opposite to longest side. Let θ be greatest angle, then cosθ=2pqp2+q2−p2−q2−pq​=−21​ ⇒ cosθ=cos32π​ ⇒ θ=32π​