Properties of Triangles

In a triangle ABC , with usual notations if $\mathrm{a}=4, \mathrm{~b}=8, \angle \mathrm{C}=60^{\circ}$, then the value of $\angle \mathrm{B}$ and the ratio $\cos \mathrm{A}: \cos \mathrm{C}$ respectively are,

With usual notation, in triangle ABC , $\mathrm{m} \angle \mathrm{A}=30^{\circ}$ then the value of $\left(1+\frac{\mathrm{a}}{\mathrm{c}}+\frac{\mathrm{b}}{\mathrm{c}}\right)\left(1+\frac{\mathrm{c}}{\mathrm{b}}-\frac{\mathrm{a}}{\mathrm{b}...

In a triangle ABC , with usual notations, $\tan \left(\frac{\mathrm{A}}{2}\right)=\frac{5}{6}, \tan \left(\frac{\mathrm{C}}{2}\right)=\frac{2}{5}$, then

With usual notations, the perimeter of a triangle ABC is 6 times the arithmetic mean of sine of its angles. If $\mathrm{a}=1$, then $\angle \mathrm{A}=$

With usual notations, in a triangle $A B C$, if $\theta$ is any real number, then $a \cos (B-\theta)+b \cos (A+\theta)$ is

In a triangle ABC with usual notations, if $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in arithmetic progression, then, $\tan \frac{\mathrm{A}}{2} \cdot \tan \frac{\mathrm{C}}{2}=$

In a triangle ABC , with usual notations if $\frac{2 \cos \mathrm{~A}}{\mathrm{a}}+\frac{\cos \mathrm{B}}{\mathrm{b}}+\frac{2 \cos \mathrm{C}}{\mathrm{c}}=\frac{\mathrm{a}}{\mathrm{bc}}+\frac{\mathrm{b}}{\mathrm{ca}}$ then $\angle \mathrm{A...

In a triangle ABC with usual notations if $|\overline{\mathrm{BC}}|=8,|\overline{\mathrm{CA}}|=7,|\overline{\mathrm{AB}}|=10$ then the projection of $\overline{\mathrm{AB}}$ on $\overline{\mathrm{AC}}$ is

In a triangle ABC , the sides $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are such that they are the roots of the equation $x^3-11 x^2+38 x-40=0$ Then
\(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\)

In a triangle $A B C$, with usual notations, $3 \mathrm{~b}=\mathrm{a}+\mathrm{c}$, then $\cot \frac{\mathrm{A}}{2} \cdot \cot \frac{\mathrm{C}}{2}=$

In a triangle ABC with usual notations if $\mathrm{a}=13$, $b=14, c=15$ Then $\sin A=$

The ratios of sides in a triangle ABC are $5: 12: 13$ and its area is 270 . Then sides of the triangle are

In a triangle $A B C$, with usual notations, if $\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}$ Then $\cos \mathrm{A}: \cos \mathrm{B}: \cos \mathrm{C}$ is

The smallest angle of the triangle whose sides are $6+\sqrt{12}, \sqrt{48}, \sqrt{24}$ is

If two sides of a triangle are $\sqrt{3}-2$ and $\sqrt{3}+2$ units and their included angle is $60^{\circ}$, then the third side of the triangle is

For the triangle ABC , with usual notations, if the angles $A, B, C$ are in A.P. and $\mathrm{m} \angle \mathrm{A}=30^{\circ}, \mathrm{c}=3$, then the values of a and b are respectively

The sides of a triangle are $\sin \theta, \cos \theta$ and $\sqrt{1+\sin \theta \cos \theta}$ for some $0<\theta<\frac{\pi}{2}$, then the greatest angle of a triangle is

If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are the angles of a triangle with $\tan \frac{A}{2}=\frac{1}{3}, \tan \frac{B}{2}=\frac{2}{3}$ then the value of $\tan \frac{C}{2}$ is

If the angles of a triangle are in the ratio $4: 1: 1$, then the ratio of the longest side to the perimeter is

In $\triangle \mathrm{ABC}$, with usual notations, if $\mathrm{b}=3$, $c=8, \mathrm{~m} \angle \mathrm{~A}=60^{\circ}$, then the circumradius of the triangle is _______ units.

If $(a+b) \cos C+(b+c) \cos A+(c+a) \cos B=72$ and if $a=18, b=24$, then area of the triangle $A B C$ is

In $\triangle A B C$, with usual notations, if $\frac{1}{b+c}+\frac{1}{c+a}=\frac{3}{a+b+c}$, then $m \angle C$ is equal to

If the lengths of the sides of triangle are 3,5,7, then the largest angle of the triangle is

In a triangle ABC , with usual notations, $2 \mathrm{ac} \sin \left(\frac{\mathrm{A}-\mathrm{B}+\mathrm{C}}{2}\right)$ is equal to

In a triangle $\mathrm{ABC}, l(\mathrm{AB})=\sqrt{23}$ units, $l(\mathrm{BC})=3$ units, $l(\mathrm{CA})=4$ units, then $\frac{\cot A+\cot C}{\cot B}$ is

In a triangle ABC , with usual notations, $\frac{\cos \mathrm{B}+\cos \mathrm{C}}{\mathrm{b}+\mathrm{c}}+\frac{\cos \mathrm{A}}{\mathrm{a}}$ has the value

If in a triangle $A B C$, with usual notations, the angles are in A.P. and $b: c=\sqrt{3}: \sqrt{2}$, then angle $\mathrm{A}=$

With usual notations, if the lengths of the sides of a triangle are $7 \mathrm{~cm}, 4 \sqrt{3} \mathrm{~cm}$ and $\sqrt{13} \mathrm{~cm}$, then the measures of the smallest angle is

If the sides of a triangle $a, b, c$ are in A.P., then with usual notations, a $\cos ^2 \frac{\mathrm{C}}{2}+\mathrm{c} \cos ^2 \frac{\mathrm{~A}}{2}$ is

In a $\triangle \mathrm{PQR}, \mathrm{m} \angle \mathrm{R}=\frac{\pi}{2}$. If $\tan \left(\frac{\mathrm{P}}{2}\right)$ and $\tan \left(\frac{\mathrm{Q}}{2}\right)$ are the roots of the equation $a x^2+b x+c=0(a \neq 0)$, then

The angles of a triangle are in the ratio $5: 1: 6$, then ratio of the smallest side to the greatest side is

If the angles $\mathrm{A}, \mathrm{B}$ and C of a triangle ABC are in the ratio $2: 3: 7$ respectively, then the sides a, b and c are respectively in the ratio

In a triangle ABC , with usual notations, if $\mathrm{m} \angle \mathrm{A}=45^{\circ}, \mathrm{m} \angle B=75^{\circ}$, then $\mathrm{a}+\mathrm{c} \sqrt{2}$ has the

Two sides of a triangle are \(\sqrt{3}+1\) and \(\sqrt{3}-1\) and the included angle is \(60^{\circ}\), then the difference of the remaining angles is

In \(\triangle \mathrm{ABC}\), with usual notations, \(\mathrm{m} \angle \mathrm{C}=\frac{\pi}{2}\), if \(\tan \left(\frac{A}{2}\right)\) and \(\tan \left(\frac{B}{2}\right)\) are the roots of the equation $$a_1 x^2+b_1 x+c_1=0\left(a_1 \ne...

Let \(a, b, c\) be the lengths of sides of triangle \(A B C\) such that \(\frac{a+b}{7}=\frac{b+c}{8}=\frac{c+a}{9}=k\). Then \(\frac{(\mathrm{A}(\triangle \mathrm{ABC}))^2}{\mathrm{k}^4}=\)

In \(\triangle \mathrm{PQR}, \sin \mathrm{P}, \sin \mathrm{Q}\) and \(\sin \mathrm{R}\) are in A.P., then

If the angles \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\) of a triangle are in an Arithmetic Progression and if \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) denote the lengths of the sides opposite to A, B and C respectively, then the v...

In \(\triangle \mathrm{ABC}\), with usual notations, \(2 \mathrm{ac} \sin \left(\frac{1}{2}(\mathrm{~A}-\mathrm{B}+\mathrm{C})\right)\) is equal to

In a triangle \(\mathrm{A B C, m \angle A, m \angle B, m \angle C}\) are in A.P. and lengths of two larger sides are 10 units, 9 units respectively, then the length (in units) of the third side is

Angles of a triangle are in the ratio \(4: 1: 1\). Then the ratio of its greatest side to its perimeter is

In \(\triangle A B C\) with usual notation, \(\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}\) and \(a=\frac{1}{\sqrt{6}}\), then the area of triangle is _______ sq. units.

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 (in units) are

In \(\triangle A B C\), with usual notations, if \(\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}\), then the value of \(\cos A+\cos B+\cos C\) is

In a triangle, the sum of lengths of two sides is \(x\) and the product of the lengths of the same two sides is \(y\). If \(x^2-\mathrm{c}^2=y\), where \(\mathrm{c}\) is the length of the third side of the triangle, then the circumradius of...

In \(\triangle \mathrm{ABC}, \mathrm{m} \angle \mathrm{B}=\frac{\pi}{3}\) and \(\mathrm{m} \angle \mathrm{C}=\frac{\pi}{4}\). Let point \(\mathrm{D}\) divide \(\mathrm{BC}\) internally in the ratio \(1: 3\), then $$\frac{\sin (\angle B A D)...

If two angles of \(\triangle \mathrm{ABC}\) are \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\), then the ratio of the smallest and greatest sides are

The lengths of sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the length of the sides of the triangle (in units) are

In a triangle \(\mathrm{ABC}\), with usual notations, if \(\mathrm{m} \angle \mathrm{A}=60^{\circ}, \mathrm{b}=8, \mathrm{a}=6\) and \(\mathrm{B}=\sin ^{-1} x\), then \(x\) has the value

If in \(\triangle \mathrm{ABC}\), with usual notations, \(a \cdot \cos ^2 \frac{C}{2}+c \cos ^2 \frac{A}{2}=\frac{3 b}{2}\), then