Embibe Experts Solutions for Chapter: Properties of Triangle, Exercise 3: Exercise-3

Given an isosceles triangle, whose one angle is $120°$ and radius of its incircle is $\sqrt{3}$. Then the area of triangle in sq. units is

Let $ABC$ be a triangle such that $\angle ACB=\frac{\pi }{6}$ and let $a, b$ and $c$ denote the lengths of the sides opposite to  $A, B$ and $C$ respectively. The value(s) of $x$ for which $a=x^2+x+1,b=x^2-1$ and $c=2x+1$ is (are)

Let $PQR$ be a triangle of area $\Delta$ with $a=2, b=\frac{7}{2}$ and $c=\frac{5}{2}$, where $a\mathit{,} b$ and $c$ are the lengths of the sides of the triangle opposite to the angles at $P, Q$ and $R$ respectively. Then $\frac{2 \sin P-\sin 2P}{2 \sin P+\sin 2P}$ equals

If in a triangle $ABC,$ $a {\cos }^2\left(\frac{C}{2}\right)+c {\cos }^2\left(\frac{A}{2}\right)=\frac{3b}{2},$ then the sides $a,b$ and $c$-

In a triangle $ABC$, medians $AD\mathit{ }$and $BE$ are drawn. If $AD=4, \angle DAB=\frac{\mathrm{\pi }}{6}$ and $\angle ABE=\frac{\mathrm{\pi }}{3}$, then the area of $\mathrm{\Delta }ABC$ is

The sides of a triangle are $\sin \alpha , \cos \alpha$ and $\sqrt{1+\sin \alpha \cos \alpha }$ for some $0<\alpha <\frac{\pi }{2}.$ Then, the greatest angle of the triangle is 

If in a $\Delta ABC,$ the altitudes from the vertices $A, B$ and $C$ on opposite sides are in $HP,$ then $\sin A, \sin B$ and $\sin C$ are in

For a regular polygon, let $r$ and $R$ be the radii of the inscribed and the circumscribed circles. A false statement among the following is