Properties of Triangles

If the angles of a triangle are in the ratio \(4: 1: 1\), then the ratio of the longest side to its perimeter is

In triangle \(\mathrm{ABC}\) with usual notations \(\mathrm{b}=\sqrt{3}, \mathrm{c}=1, \mathrm{~m} \angle \mathrm{A}=30^{\circ}\), then the largest angle of the triangle is

If the vertices of a triangle are \((-2,3),(6,-1)\) and \((4,3)\), then the co-ordinates of the circumcentre of the triangle are

If one side of a triangle is double the other and the angles opposite to these sides differ by \(60^{\circ}\), then the triangle is

In a triangle \(\mathrm{ABC}\), with usual notations, if \(\mathrm{c}=4\), then value of \((a-b)^2 \cos ^2 \frac{C}{2}+(a+b)^2 \sin ^2 \frac{C}{2}\) is

In any \(\triangle A B C\), with usual notations, \(c(a \cos B-b \cos A)=\)

If in a \(\triangle A B C\), with usual notations, \(\mathrm{a}^2, \mathrm{~b}^2, \mathrm{c}^2\) are in A.P. then \(\frac{\sin 3 B}{\sin B}=\)

In a triangle ABC with usual notations a = 2, b = 3, then value of \(\frac{\cos 2 \mathrm{~A}}{\mathrm{a}^2}-\frac{\cos 2 \mathrm{~B}}{\mathrm{~b}^2}\) is

In a triangle \(\mathrm{ABC}\), with usual notations \(\mathrm{a}=2, \mathrm{~b}=3, \mathrm{c}=5\), then \(\frac{\cos \mathrm{A}}{\mathrm{a}}+\frac{\cos \mathrm{B}}{\mathrm{b}}+\frac{\cos \mathrm{C}}{\mathrm{c}}=\)

With usual notations in \(\triangle\)ABC, if \(\frac{\sin A}{\sin C}=\frac{\sin (A-B)}{\sin (B-C)}\), then \(a^2, b^2, c^2\) are in

If \(\mathrm{G}(\overline{\mathrm{g}}), \mathrm{H}(\overline{\mathrm{h}})\) and \(\mathrm{P}(\overline{\mathrm{p}})\) are respectively centroid, orthocenter and circumcentre of a triangle and $$\mathrm{x} \overline{\mathrm{p}}+\mathrm{y} \o...

The area of the triangle \(\mathrm{ABC}\) is \(10 \sqrt{3} \mathrm{~cm}^2\), angle \(\mathrm{B}\) is \(60^{\circ}\) and its perimeter is \(20 \mathrm{~cm}\), then \(\ell(\mathrm{AC})=\)

If in \(\Delta\)ABC, with usual notations, the angles are in A.P., then \(\mathrm{\frac{a}{c}}\) sin 2 C + \(\mathrm{\frac{c}{a}}\) sin 2 A =

In \(\triangle A B C\), with usual notations, \(2 a b \sin \frac{1}{2}(A+B-C)=\)

With usual notations, in any \(\triangle A B C\), if \(a\cos B=b \cos A\), then the triangle is

In \(\Delta ABC\), with usual notations \(\mathrm{\frac{b\sin B-c\sin C}{\sin(B-C)}}=\)

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

With usual notations if the angles of a triangle are in the ratio 1 : 2 : 3, then their corresponding sides are in the ratio.

The area of the $\triangle A B C$ is $10 \sqrt{3} \mathrm{~cm}^2$, angle $B$ is $60^{\circ}$ and its perimeter is 20 cm , then $\ell(A C)=$

With usual notations, if the angles $A, B, C$ of a $\triangle A B C$ are in $A P$ and $b: c=\sqrt{3}: \sqrt{2}$

In a triangle \(A B C\), if \(\frac{\sin A-\sin C}{\cos C-\cos A}=\cot B\), then \(A, B, C\), are in

In a triangle \(A B C\) with usual notations, if \(\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}\), then area of triangle \(A B C\) with \(a=\sqrt{6}\) is

In a \(\triangle A B C\) if \(2 \cos C=\sin B \cdot \operatorname{cosec} A\), then

In $\triangle A B C$, with the usual notations, if $\sin B \sin C=\frac{b c}{a^2}$, then the triangle is. ...........

In $\triangle A B C$, with the usual notations, if $\left(\tan \frac{A}{2}\right)\left(\tan \frac{B}{2}\right)=\frac{3}{4}$ then $a+b=\ldots \ldots$

In $\triangle A B C$; with usual notations, if $\cos A=\frac{\sin B}{\sin C}$ then the triangle is ............

In $\triangle A B C$; with usual notations,
\(\frac{b \sin B-c \sin C}{\sin (B-C)}=\ldots \ldots\)

In $\triangle A B C$, if $\tan A+\tan B+\tan C=6$ and $\tan A \cdot \tan B=2$ then $\tan C=$ ...........

If $R$ is the circum radius of $\triangle A B C$, then $A(\triangle A B C)=\ldots \ldots$