Properties of triangles practice · Mixed difficulty

If ${A_1}\,{A_2}\,{A_3}\,{A_4}\,{A_5}$ be a regular pentagon inscribed in a unit circle. Then $\left( {{A_1}{A_2}} \right)\,\,\left( {{A_1}{A_3}} \right)$ is equal to

In the ambiguous case, given $a,\,\,b$ and $A$. The difference between the two values of $C$ is

In a $\triangle A B C, a=2 b$ and $|A-B|=\frac{\pi}{3}$. The measure of $\angle C$ is

If $a, b$ and $A$ are given in $a$ triangle and $c_1, c_2$ are the possible values of the third side, then $c_1^2+c_2^2-2 c_1 c_2 \cos 2 A$ is equal to

The angles $A,\;B$ and $C$ of a $\Delta \;ABC$ are in AP. if $b:c = \sqrt 3 :\sqrt 2$, then the angle $A$ is

In an equilateral triangle, the ratio of the inradius, circumradius and exradius are

In a triangle $A B C$, let $\angle C=\frac{\pi}{2}$. If $r$ is the inradius and $R$ is the circumradius of the triangle $A B C$, then $2(r+R)$ equals

If in a triangle $ABC,\,\,\cos A\,\,\cos B + \sin A\,\,\sin \,B\,\,\sin C = 1$, then the triangle is

If $\alpha, \beta$ and $\gamma$ are the altitudes of a $\triangle A B C$ and $2 s$ denotes its perimeter, then $\alpha^{-1}+\beta^{-1}+\gamma^{-1}$ is equal to

In the angles $A,B$ and $C$ of a triangular are in the arithmetic progression and if $a,b$ and $c$ denotes the lengths of the sides opposite to $A,B$ and $C$ respectively, then the value of the expression $\frac{a}{c}\sin \;2C + \frac{c}{a}\sin \;2A$ is

If in a $\triangle A B C, a^2+b^2+c^2=8 R^2$, where $R=$ circumradius, the the triangle is

If in a triangle $ABC,\,\,b + c = 3a$, then $\tan \,\left( {\frac{B}{2}} \right)\tan \left( {\frac{C}{2}} \right)$ is equal to

Angles $A,B$ and $C$ of a $\Delta \;ABC$ are in AP. If $\frac{b}{c} = \frac{{\sqrt 3 }}{{\sqrt 2 }}$, then angle $A$ is equal to

If in a $\Delta ABC,\;{r_1} < {r_2} < {r_3}$, then

$\sin A:\sin C = \sin \left( {A - B} \right):\sin \left( {B - C} \right)$, then ${a^2},\;{b^2},\;{c^2}$ are in

In a $\Delta \,ABC$, if $c = 2,\,\,A = 120^\circ ,\,\,a = \sqrt 6$, then $C =$

If $P$ is a point on the altitude $A D$ of the $\triangle A B C$ such that $\angle C D P=\frac{B}{3}$, then $A P$ is equal to

If the angles $A, B$ and $C$ of triangle $A B C$ are in arithmetic progression and $a, b, c$ represents length of sides opposite to angles $A, B$ and $C$ respectively, then the value of $\frac{a+c}{\sqrt{\left(a^2-a c+c^2\right)}}$ is:

In $\Delta\; ABC$, if ${\sin ^2}\frac{A}{2},{\sin ^2}\frac{B}{2},{\sin ^2}\frac{C}{2}$ be in HP, then $a,\;b,\;c$ will be in

If $a, b$ and $c$ are the sides of a triangle such that $b \cdot c=\lambda^2$, for some positive $\lambda$, then

The area of the segment of a circle of radius $a$ subtending an angle of $2\alpha$ at the centre is

If in a $\Delta ABC,\;4\sin A = 4\sin B = 3\sin C$, then $\cos C$ is equal to

In a $\Delta \,ABC,\,\,A = \frac{{2\;\pi }}{3},\,\,b - c = 3\,\sqrt 3 \;\mathrm{cm}$ and $\Delta = \frac{{9\,\sqrt 3 }}{2}\,\,\mathrm{c}{\mathrm{m}^2}$. Then, $a =$

In a $\triangle A B C, \tan \frac{A}{2}=\frac{5}{6}, \tan \frac{C}{2}=\frac{2}{5}$, then

In a triangle ${r_1} > {r_2} > {r_3}$, then

In any triangle $\mathrm{ABC}$, $\mathrm{a} \cot \mathrm{A}+\mathrm{b} \cot \mathrm{B}+\mathrm{c} \cot \mathrm{C}=$

If the data given to construct a triangle $ABC$ are $a = 5,\,\,b = 7,\,\,\sin A = 3/4$, then it is possible to construct

The longest side of a triangle is 3 times the shortest side and the third side is $2 \mathrm{~cm}$ shorter than the longest side. If the perimeter of the triangle is at least $61 \mathrm{~cm}$, find the minimum length of the shortest side.

In $\Delta ABC,2\left( {a\;{{\sin }^2}\frac{C}{2} + c\;{{\sin }^2}\frac{A}{2}} \right)$ is equal to

If ${c^2} = {a^2} + {b^2},\,\,2\;s = a + b + c$, then $4s\,\,\left( {s - a} \right)\,\,\left( {s - b} \right)\,\,\left( {s - c} \right) =$