Properties of triangles practice · Easy

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 $\alpha, \beta$ 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 $ABC,\,\,b + c = 3a$, 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

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

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

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 $4s\,\,\left( {s - a} \right)\,\,\left( {s - b} \right)\,\,\left( {s - c} \right) =$

In $\triangle A B C, 1-\tan \frac{B}{2} \cdot \tan \frac{C}{2}$ is equal to

If $r_1=2 r_2=3 r_3$ then $a+b+c$ is equal to

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

A quadrilateral $ABCD$ in which $AB = a,\;BC = b,\;CD = c$ and $DA = d$ is such that one circle can be inscribed in it and another circle can be circumscribed about it, then $\cos A$ is equal to

In a $\triangle A B C$, if $A=60^{\circ}$, then $\frac{b}{c+a}+\frac{c}{a+b}$ is equal to

In a $\Delta \;ABC,\;a,\;c\;,\;A$ are given and ${b_1},{b_2}$ are two values of third side $b$ such that $b = 2{b_1}$. Then, $\sin A$ is equal to

In a $\Delta \,ABC$,

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

In a $\Delta \,ABC$ if $c = \left( {a + b} \right)\sin \,\,\theta$ and $\cos \,\,\theta = \frac{{k\,\sqrt {ab} }}{{a + b}}$, then $k =$

If $A = 30^\circ ,\,\,a = 7,\,\,b = 8$ in $\Delta \,ABC$, then $B$ has

If a $\Delta \;ABC,\mathrm{if}\;{a^4} + {b^4} + {c^4} = 2{c^2}\left( {{a^2} + {b^2}} \right)$, then $\angle C$ is equal to

The horizontal distance between two tower is $60 \mathrm{~m}$. The angular elevation of the top of the taller tower as seen from the top of the shorter one is $30^{\circ}$. If the height of the taller tower is $150 \mathrm{~m}$, the height of the shorter one is

If ${p_1},\,\,{p_2},\,\,{p_3}$ are altitudes of a triangle $ABC$ from the vertices $A,\,\,B,\,\,C$ and $\Delta$, the area of the triangle, then $p_1^{ - 1} + p_2^{ - 1} - p_3^{ - 1}$ is equal to