Properties of triangles practice · Mixed difficulty

In a $\Delta ABC$, $\frac{{\left( {a + b + c} \right)\left( {b + c - a} \right)\left( {c + a - b} \right)\left( {a + b - c} \right)}}{{4{b^2}{c^2}}}$ equals

In a triangle $ABC,\frac{{\cos A}}{a} = \frac{{\cos B}}{b} = \frac{{\cos C}}{c}$. if $a = \frac{1}{{\sqrt 6 }}$, then the area of the triangle (in square unit) is

Consider a $\triangle A B C$ and let $a, b$ and $c$ denote the lengths of the sides opposite to vertices $A, B$ and $C$, respectively. Suppose $a=6, b=10$ and the area of the triangle is $15 \sqrt{3}$. If $\angle A C B$ is obtuse and if $r$ denotes the radius of the incircle of the triangle, then $r^2$ is equal to

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

If $\Delta ABC,\mathrm{if}a = 3,b = 4,c = 5$, then the value of $\sin 2B$ is

If the sides of a triangle are ${x^2} + x + 1,\,\,{x^2} - 1,\,\,2x + 1$, where $x > 1$, then the largest angle is

In a $\Delta \,ABC$, if $B = 90^\circ$, then the value of $\tan \frac{A}{2}$ in terms of the sides is

If in a $\triangle A B C, a \tan A+b \tan B=(a+b) \tan \frac{(A+B)}{2}$, then

In $\Delta \;ABC,\left( {a + b + c} \right)\left( {\tan \frac{A}{2} + \tan \frac{B}{2}} \right)$ is equal to

Three poles whose feet lie on a circle subtend angle $\alpha, \beta, \gamma$ respectively at the centre of the circle. If the height of the poles are in A.P. then $\cot \alpha, \cot \beta, \cot \gamma$ are in

If $A,\;{A_1},{A_2},\;{A_3}$ be the areas of the incircle and excircles, then $\frac{1}{{\sqrt {{A_1}} }} + \frac{1}{{\sqrt {{A_2}} }} + \frac{1}{{\sqrt {{A_3}} }}$ is equal to

In a $\triangle A B C, 2 c a \sin \left(\frac{A-B+C}{2}\right)$ is equal to

If $a = 2\sqrt 2 ,\;b = 6,\;A = 45^\circ$, then

In a $\Delta ABC$, if $2s = a + b + c$ and $\left( {s - b} \right)\left( {s - c} \right) = x\;{\sin ^2}\frac{A}{2}$, then the value of $x$ is

The radii $r_1, r_2, r_3$ of escribed circles of the $\triangle A B C$ are in HP. If its area is $24 \mathrm{sq} \mathrm{cm}$ and its perimeter is $24 \mathrm{~cm}$, then the lengths of its sides are

In a triangle $A B C, \tan \frac{A-B}{2} \cot \frac{A+B}{2}$ is equal to

If in $\Delta \,\;ABC,\;\,a\,\,\sin \,\,A = b\,\,\sin \,\,B$, then the triangle is

In a triangle $ABC,\,\,\cos \,\,A + \cos \,\,B + \cos \,\,C =$

In a $\mathrm{\Delta }\,ABC,\,\,a = 2b$ and $A = 3B$, the $A =$

If in a $\triangle A B C$, $\cos A \cos B+\sin A \sin B \sin C=1$, then $a: b: c$ is equal to

The area of the triangle inscribed in a circle of radius of 4 and the measures of whose angles are in the ratio $5:4:3$ is

In a $\triangle A B C, \frac{\cos C+\cos A}{c+a}+\frac{\cos B}{b}$ is equal to

In a $\Delta ABC,\left( {a + b + c} \right)\left( {b + c - a} \right) = kbc$, if

In a $\triangle A B C, a=8 \mathrm{~cm}, b=10 \mathrm{~cm}, c=12 \mathrm{~cm}$. Then, relation between angles of the triangle is

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then, the sides of the triangle are

In a $\triangle A B C, a, b, c, A$ are given and $c_1, c_2$ are two values of the third side $c$. The sum of the areas of two triangles with sides $a, b, c_1$ and $a, b, c_2$ is

In a $\Delta \,ABC$ if $a = 26,\,\,b = 30$ and $\cos \,\,C = \frac{{63}}{{65}}$, then ${r_2} =$

In a $\Delta \;ABC$, if $a,\;b,\;c$ are in AP, then the value of $\frac{{\sin \frac{A}{2}\sin \frac{C}{2}}}{{\sin \frac{B}{2}}}$ is

In triangle ABC, the value of $\frac{{\cot \frac{A}{2}\cot \frac{B}{2} - 1}}{{\cot \frac{A}{2}\cot \frac{B}{2}}}$ is

The angle of a right angled triangle are in A.P. The ratio of the in-radius and the perimeter is