Properties of triangles practice · Mixed difficulty

In a triangle $A B C a=6, b=3 \& \cos (A-B)=4 / 5$. Find area of $\Delta$.

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

If the angles of a $\Delta \;ABC$ be in AP, then

In a $\Delta \,ABC,\,\,\frac{{\cot \,\,\frac{A}{2} + \cot \,\,\frac{B}{2} + \cot \,\,\frac{C}{2}}}{{\cot \,\,A + \cot \,\,B + \cot \,\,C}} =$

If in a triangle, $R$ and $r$ are the circumradius and inradius respectively, then the HM of the exradii of the triangle is

If the angle $\mathrm{A}$ of a triangle $\mathrm{ABC}$ is given by the equation $5 \cos A+3=0$, then $\sin A$ and $\tan A$ are the roots of the equation

In triangle $ABC,\,\,A = 30^\circ ,\,\,b = 8,\,\,a = 6$, then $B = {\sin ^{ - 1}}x$, where $x =$

In a $\triangle \mathrm{ABC}$, if angle $\mathrm{C}$ is obtuse, then

In a $\Delta \;ABC,\;\mathrm{if}\;{r_1} = 2{r_2} = 3{r_3}$, then

In any triangle $ABC,\;{c^2}\sin 2B + {b^2}\sin 2C$ is equal to

In a $\Delta \,ABC,\,\,b = 2,\,\,C = 60^\circ ,\,\,c = \sqrt 6$, then $a =$

The inradius of the triangle whose sides are 3, 5, 6 is

If $A, B$ and $C$ are angles of triangle such that $\angle A$ is obtuse, then $\tan B \tan C$ will be less than

If $R$ and $R^{\prime}$ are the resultant of two forces $\frac{P}{Q}$ and $\frac{{Q}}{{P}}({P}>{Q})$ according as they are like or unlike such that ${R}: {R}^{\prime}=25: 7$, then ${P}: {Q}=$

In any $\triangle A B C, 2 a \cos \frac{B}{2} \cos \frac{C}{2}$ is equal to

In a triangle vertex angles are $A,\;B,\;C$ and side $BC$ are given. The area of $\Delta ABC$ is

If the lengths of the sides of a triangle are in AP and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is

The area of the triangle whose sides are $6,\;5,\;\sqrt {13}$ (in square unit) is

If the altitudes of a triangle are in AP, then the sides of the triangle are in

In a right-angled triangle, the sides are $a,\;b$ and $c$, with $c$ as hypotenuse, and $c - b \ne 1,\;c + b \ne 1$. Then the value of $\left( {{{\log }_{c + b}}a + {{\log }_{c - b}}a} \right)/\left( {2{{\log }_{c + b}}a \times {{\log }_{c - b}}a} \right)$ will be

If the radius of the incircle of a triangle with its sides $5\,k,\,\,6\,k$, and $5k$ is $\mathrm{6}$, then $k$ is equal to

The area of a triangle $A B C$, where $a=2(\sqrt{3}+1), B=45^{\circ}, C=60^{\circ}$ is

In a $\Delta ABC,a:b:c = 4:5:6$. The ratio of the radius of the circumcircle to that of the incircle is

If ${a^2},{b^2},{c^2}$ are in AP, then which of the following are also in AP?

In a triangle $A B C$, if $\left(1+\frac{a}{b}+\frac{c}{b}\right)\left(1+\frac{b}{c}-\frac{a}{c}\right)=3$, then the angle $A$ is equal to

If ${\cos ^2}A + {\cos ^2}C = {\sin ^2}B$, then $\Delta ABC$ is

The radius of the incircle of triangle when sides are 18, 24 and 30 cm is

In a triangle, the sum of two sides is $x$ and the product of the same two sides is $y$. If $x^2-c^2=y$, where $c$ is the third side of the triangle, then the ratio of the inradius to the circumradius of the triangle is

A triangle $A B C$ is inscribed in a circle of radius $4 \mathrm{~cm} . \angle A=60^{\circ} A C-A B=4 \mathrm{~cm}$, then area of $\triangle \mathrm{ABC}$ will be?

In any $\triangle A B C, r r_1+r_2 r_3$ is equal to