Properties of triangles practice · Mixed difficulty

In $\Delta \;ABC,\frac{{1 + \cos \left( {A - B} \right)\cos C}}{{1 + \cos \left( {A - C} \right)\cos B}}$ is equal to

In any $\triangle A B C$, if $C=90^{\circ}$, then $\tan \frac{B}{2}$ is equal to

In a $\Delta \,ABC$, if $a = 8,\,\,b = 10$ and $c = 12$, then $C$ is equal to

Let $a, b$ and $c$ be real numbers such that $4 a+2 b+c$ $=0$ and $a b>0$. Then the quadratic equation $a x^2+b x$ $+c=0$ has

If the sides of a triangle are the roots of the equation ${x^3} - 2{x^2} - x - 16 = 0$, then the product of the in-radius and circum-radius of the triangle is

If $A=30^{\circ}, a=100, c=100 \sqrt{2}$, find the number of triangles that can be formed.

The smallest angle of the triangle whose sides are $6 + \sqrt {12} ,\,\sqrt {48} ,\sqrt {24}$ is

If a $\Delta ABC,\;2ca\sin \frac{{A - B + C}}{2}$ is equal to

In a triangle with one angle if $120^\circ$, the length of the sides forms an AP. If the length of the greatest sides is 7 cm, then area of triangle is

If in a $\Delta \;ABC,\cot \frac{A}{2} = \frac{{b + c}}{a}$, then the $\Delta \;ABC$ is

In a $\Delta \,ABC,\,\,{\cos ^2}\,\,\frac{A}{2}\,\, + \,\,{\cos ^2}\,\,\frac{B}{2}\, + \,{\cos ^2}\,\,\frac{C}{2} =$

In a $\Delta \;ABC$, if $\frac{{\cos A}}{a} = \frac{{\cos B}}{b} = \frac{{\cos C}}{c}$ and the side $a = 2$, then area of the triangle is

If $R$ is the radius of the circumcircle of the $\triangle A B C$ and its area is $\Delta$, then

The area of the circle and the area of a regular polygon of $n$ sides and of perimeter equal to that of the circle are in the ratio of

If in a $\triangle A B C, r_1=2, r_2=3$ and $r_3=6$, then $a$ equals

If sides of a triangle are $18,24,30 \mathrm{~cm}$, then radius of circumcircle is

In a ∆ABC, the correct formulae among the following are

III. $\;\;r = 4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

IV. $\;\;{r_1} = \left( {s - a} \right)\tan \frac{A}{2}$

V. $\;\;{r_3} = \frac{\Delta }{{s - c}}$

If, in a $\Delta \,ABC,\,\,\left( {a + b + c} \right)\,\,\left( {b + c - a} \right) = \lambda \;\,bc$, then

In a triangle $\mathrm{ABC}, 2 a^2+4 b^2+c^2=4 a b+2 a c$, then $\cos \mathrm{B}$ is equal to

Angles $A,B$ and $C$ of a triangle are in AP with common difference 15 degree, then angle $A$ is equal to

In a $\Delta ABC$, if $\tan \frac{A}{2} = \frac{5}{6},\tan \frac{C}{2} = \frac{2}{5}$, then

In $\Delta \;ABC,{a^2}\left( {{{\cos }^2}B - {{\cos }^2}C} \right) + {b^2}\left( {{{\cos }^2}C - {{\cos }^2}A} \right) + {c^2}\left( {{{\cos }^2}A - {{\cos }^2}B} \right)$ is equal to

In a $\Delta \,ABC,\,\,a = 5,\,\,b = 4$ and $\cos \left( {A - B} \right) = \frac{{31}}{{32}}$, then side $c$ is

In a $\Delta \,ABC$, angles $A,\,\,B,\,\,C$ are in A.P., then
$\lim\limits_{A \to C} \,\,\frac{{\sqrt {3 - 4\,\,\sin \,\,A\,\,\sin C} }}{{\left| {\,A - C\,} \right|}}$ is equal to

In a $\Delta \;ABC,\angle \mathrm{ }C = 60^\circ$ then $\frac{1}{{a + c}} + \frac{1}{{b + c}}$ is equal to

In $\triangle \mathrm{ABC}$, if $\mathrm{a}: \mathrm{b}: \mathrm{c}=1: \sqrt{3}: 2$, then the ratio $\mathrm{A}: \mathrm{B}: \mathrm{C}$ is equal to

If the area of a $\Delta \;ABC$ is given by $\Delta = {a^2} - {\left( {b - c} \right)^2}$, then $\tan \left( {\frac{A}{2}} \right)$ is equal to

In any $\triangle A B C$, the simplified form of $\frac{\cos 2 A}{a^2}-\frac{\cos 2 B}{b^2}$ is

The angles of elevation of a vertical tower standing inside a triangular field at the vertices of the field are each equal to $\theta$. If the length of the sides of the field are $30 \mathrm{~m}, 50 \mathrm{~m}$ and $70 \mathrm{~m}$, the height of the tower is

If in the $\Delta \;ABC,\;\angle B = 45^\circ$, then ${a^4} + {b^4} + {c^4}$ is equal to