Properties of triangles practice · Easy

If $a,\;b,\;c$ be the sides of a $\Delta \;ABC$ and if roots of the equation $a\left( {b - c} \right){x^2} + b\left( {c - a} \right)x + c\left( {a - b} \right) = 0$ are equal, then ${\sin ^2}\left( {\frac{A}{2}} \right),{\sin ^2}\left( {\frac{B}{2}} \right),{\sin ^2}\left( {\frac{C}{2}} \right)$ are in

In $\Delta \,ABC$, if $\frac{{s - a}}{\Delta } = \frac{1}{8},\,\,\frac{{s - b}}{\Delta } = \frac{1}{{12}}$ and $\frac{{s - c}}{\Delta } = \frac{1}{{24}}$, then $b =$

Given $b=2, c=\sqrt{3}, \angle A=30^{\circ}$. Then the in-radius of $\triangle A B C$ is

In a $\Delta \;ABC,\left( {b + c} \right)\left( {bc} \right)\cos A + \left( {a + c} \right)\left( {ac} \right)\cos B + \left( {a + b} \right)\left( {ab} \right)\cos C$ is

If the area of a $\triangle A B C$ be $\lambda$, then $a^2 \sin 2 B+b^2 \sin 2 A$ is equal to

In any $\Delta ABC$ under usual notation, $a\left( {b\cos C - c\cos B} \right)$ is equal to

If the sides $a,\,\,b$ and $c$ of a $\Delta \,ABC$ are in A.P., then
$\left( {\tan \,\,\frac{A}{2} + \tan \,\,\frac{C}{2}} \right):\cot \,\,\frac{B}{2}$

A $\triangle A B C$ is right angled at $B$. Then, the diameter of the incircle of the triangle is

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

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

If in a $\Delta ABC,\;a = 6$ cm, $b = 8$ cm, $c = 10$ cm, then the value of $\sin 2A$ is

The circumradius of the triangle whose sides are 13, 12 and 5, is

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

The greatest angle of a triangle whose sides are $x^2+x+1,2 x+1$ and $x^2-1$, is

If the angles of a triangle are in the ratio $1\;:2\;:3$, the corresponding sides are in the ratio

In a $\Delta ABC,a\cot A + b\cot B + c\cot C$ is equal to

In $\Delta \;ABC,\mathrm{if}\left( {a + b + c} \right)\left( {a - b + c} \right) = 3ac$, then

In a $\Delta \;ABC$, if ${b^2} + {c^2} = 3{a^2}$, then $\cot B + \cot C - \cot A$ is equal to

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

The vertices of a triangle are $A(1,1), B(4,5)$ and $C(6,13)$. Then, the value of $\cos A$ is

The ex-radii of a triangle ${r_1},\,\,{r_2},\,\,{r_3}$ are in harmonic progression, then the sides $a,\,\,b,\,\,c$ are

If $x,y,z$ are perpendicular drawn from the vertices of triangle having sides $a,b$ and $c$, then the value of $\frac{{bx}}{c} + \frac{{cy}}{a} + \frac{{az}}{b}$ will be

In a $\triangle A B C, R=$ circumradius and $r=$ inradius. The value of $\frac{a \cos A+b \cos B+c \cos C}{a+b+c}$ is equal to

In a $\Delta \,ABC$, if $a,\,\,c,\,\,b$ are in A.P., then the value of $\frac{{a\,\,\cos \,\,B - b\,\,\cos \,\,A}}{{a - b}}$, is

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

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