Properties of Triangles

In $\triangle A B C$, if $B C$ is the hypotenuse, then $r_2+r_3=$

In a $\triangle A B C$, if $(b+c)^2 \sin ^2 \frac{A}{2}+(b-c)^2 \cos ^2 \frac{A}{2}=K(1-\cos 2 A)$, then $K=$

In a $\triangle A B C$, if $b=7, c=4 \sqrt{3}$ and $A=\frac{\pi}{6}$ then a $\sin B \sin C=$

Let $A$ be the area of in-circle and $A_1, A_2, A_3$ be the area of ex-circles of a triangle. If $A_1=4, A_2=9, A_3=16$, then $A=$

In $\triangle A B C$, if $A$ is acute, $C$ is obtuse, $\sin A=\frac{3 \sqrt{3}}{14}, a=3$ and $b=5$, then $c=$

If $\Delta$ denotes the area of $\triangle A B C$, then $(b \sin C+c \sin B)(b \cos C+c \cos B)=$

In $\triangle A B C$ if $\angle C=\frac{\pi}{2}$ then
$\tan ^{-1}\left(\frac{a}{b+c}\right)+\tan ^{-1}\left(\frac{b}{c+a}\right)+\tan ^{-1}\left(\frac{c}{a+b}\right)=$

In $\triangle A B C$, if $R=\frac{65}{8}, r r_1=42$ and $r_1-r=6.5$, then $s(s-a)=$

In a $\triangle A B C$, if $\tan A: \tan B: \tan C=1: 2: 3$ and $\sin A: \sin B: \sin C=\sqrt{5}: 2 \sqrt{2}: k$, then $k=$

If the sides of a triangle are in the ratio $\sqrt{3}: \sqrt{5}: \sqrt{8+\sqrt{15}}$, then the largest angle in that triangle is

If the radius of the incircle of a triangle with sides $5 k, 6 k$ and $5 k$ is 6 , then the largest angle of that triangle is

In $\triangle A B C, A D$ and $B E$ are medians drawn from $A$ and $B$. If $A D=\frac{7}{2}, \angle D A B=\frac{\pi}{8}$ and $\angle A B E=\frac{\pi}{4}$, then the area (in sq. units) of $\triangle A B C$ is

If the sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one, then the area (in sq. units) of that triangle is

In a $\triangle A B C$, let $a, b, c, s, r, R, I, S, r_1, r_2, r_3$ stand for their usual meaning. Then Match the items of List-I with those of the items of List-II.






List-I

List-II





A.

tan

⁡

A

2


...

In a $\triangle A B C$, with usual notation, if $r=r_1-r_2-r_3$, then $2 R=$

The perimeter of a $\triangle A B C$ is 6 times the arithmetic mean of the sine of its angles. If its side $B C$ is of unit length, then $\angle A=$

In a $\triangle A B C, \cot A+\cot B+\cot C=$

In a triangle $A B C$, if $c=9, s=10$ and $\Delta=10 \sqrt{2}$ then $b\left[1+\sqrt{2} \tan \left(\frac{A-B}{2}\right)\right]=$

In a triangle $A B C$, if $a

In a $\triangle A B C$ if $\angle A=3 \angle B, C A=9$ and $B C=16$, then the length of $A B$ is

In a triangle $A B C$, if $\cos A \cos B+\sin A \sin B \sin C=1$, then $a: b: c=$

In $\triangle A B C, \frac{1+\cos C}{r_1+r_2}+\frac{1+\cos A}{r_2+r_3}+\frac{1+\cos B}{r_1+r_3}=$

If $R: r_1: r=5: 12: 2$, then $r+r_3+r_2-r_1=$

In a $\triangle A B C, \frac{\Delta^2}{a^2+b^2+c^2}\left(\frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}+\frac{1}{r^2}\right)=$

In a $\triangle A B C,\left(b^2-c^2\right) \cot A+\left(c^2-a^2\right) \cot B=$