Properties of triangles practice · Mixed difficulty

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 $c$

In $a \triangle A B C, a=1$ and the perimeter is six times the $A M$ of the sines of the angles. The measure of $\angle A$ is

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

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

If the area of $\triangle A B C$ is $a^2-(b-c)^2$, then find the value of $\tan A$.

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

For a triangle $A B C$, the value of $\cos 2 A+\cos 2 B+$ $\cos 2 C$ is least. If its inradius is 3 and incentre is $M$, then which of the following is NOT correct?

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

In a $\Delta ABC$, if $\left( {\sqrt 3 - 1} \right)a = 2b,\;A = 3B$, then $\angle C$ is

In a triangle $A B C, 2 a^2+4 b^2+c^2=4 a b+2 a c$, then the numerical value of $\cos B$ is equal to

In $\Delta \;ABC,\;2{R^2}\sin A\sin B\sin C$ is equal to

In a $\triangle P Q R$, If $3 \sin P+4 \cos Q=6$ and $4 \sin Q+3 \cos$ $P=1$, then the angle $R$ is equal to:

If in a $\Delta ABC,\left( {s - a} \right)\left( {s - b} \right) = s\left( {s - c} \right)$, then angle $C$ is equal to

In a $\Delta \;ABC$, if $a = 2x,\;b = 2y$ and $\angle C = 120^\circ$, then the area of the triangle is

In a $\Delta \;ABC$, if $\sin A + \sin B + \sin C)\left( {\sin A + \sin B - \sin C} \right) = 3\sin A\sin B$, then the angle $C$ is equal to

If $a^2+b^2+c^2=1$, then $a b+b c+c a$ lies in the interval :

The sides of two angles of a triangle are equal to $\frac{5}{13}$ and $\frac{99}{101}$. The cosine of the third angle is

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

If in a triangle $ABC,\,\,\frac{{b + c}}{{11}} = \frac{{c + a}}{{12}} = \frac{{a + b}}{{13}}$ then $\cos \,\,A$ is equal to

If the angles of a triangle be in the ratio 1:2:7, then the ratio of its greatest side to the least side is

If in a $\triangle A B C, C D$ is the angle bisector of the $\angle A B C$, then $C D$ is equal to

The angles $A,B$ and $C$ of a $\Delta \;ABC$ are in A.P. If $AB = 6,BC = 7,$ then $AC =$