Properties of Triangles

Match the items of List-I with those of List-II (Here $\Delta$ denotes the area of $\triangle A B C$.) Then the correct match is

In a triangle, if the ex-radii $r_1, r_2, r_3$ are in the ratio $1: 2: 3$, then its sides are in the ratio

If $\alpha, \beta, \gamma$ are the lengths of the tangents from the vertices of a triangle to its incircle. Then

In $\triangle A B C$, if $a: b: c=4: 5: 6$, then the ratio between the circumradius and the inradius is

In a $\triangle A B C$, let $\angle C=\frac{\pi}{2}$. If $r$ and $R$ are respectively inradius and circumradius of $A B C$, then $R+r=$

If the area of the circum-circle of triangle formed by the line $2 x+5 y+\alpha=0$ and the positive coordinate axes is $\frac{29 \pi}{4}$ sq.units, then $|\alpha|=$

In a $\(\triangle A B C\), with usual notation, match the items in List-I with the items in List-II and choose the correct option.

In a \(\triangle A B C, \angle C=60^{\circ}\) and \(\angle A=75^{\circ}\). If \(D\) is a point on \(A C\) such that the area of \(\triangle B A D\) is \(\sqrt{3}\) times the area of \(\triangle B C D\), then the measure…

The area (in square units) of $\triangle A B C$ if $\angle A=75^{\circ}, \angle B=45^{\circ}$ and $a=2(\sqrt{3}+1)$ is

If 2 is the length of a side of a triangle with its opposite angle ' $\pi / 3$ ', then the circumradius of the triangle is ......

If a circle is inscribed in an equilateral triangle of side a, then the area of any square (in sq. units) inscribed in this circle is

In a $\triangle A B C$, if $3 a=b+c$, then $\cot \frac{B}{2} \cot \frac{C}{2}=$