Embibe Experts Solutions for Chapter: Properties of Triangle, Exercise 2: Exercise-2

In a triangle $ABC,$ with usual notations the length of the bisector of internal angle $A$ is-

Let $f, g, h$ be the lengths of the perpendiculars from the circum-centre of the $△ABC$ on the sides $BC, CA$ and $AB$ respectively. If $\frac{a}{f}+\frac{b}{g}+\frac{c}{h}=\lambda \frac{abc}{fgh},$ then the value of ${}^{\text{'}}{\lambda }^{\text{'}}$ is-

If $\text{'}l\text{'}$ is the length of median from the vertex $A$ to the side $BC$ of a $△ABC,$ then

The product of the distances of the incentre from the angular points of a $△ABC$ is-

In a triangle $ABC, B=60°$ and $C=45°.$ Let $D$ divides $BC$ internally in the ratio $1 : 3,$ then value of $\frac{\sin \angle BAD}{\sin \angle CAD}$ is

In a triangle $ABC,$ points $D$ and $E$ are taken on side $BC$ such that $BD=DE=EC.$ If angle $ADE=$ angle $AED=\theta ,$ then-

STATEMENT-$1$: If $R$ be the circumradius of a $∆ABC,$ then the circumradius of its excentral $\mathrm{\Delta }{I}_1{I}_2{I}_3$ is $2R.$

STATEMENT-$2$: If the circumradius of a triangle be $R,$ then the circumradius of its pedal triangle is $\frac{R}{2}.$

Parallelogram $ABCD$ is cut by $\left(2n-1\right)$ number of parallel lines in which one is diagonal $AC.$ Distance between any two nearest lines is same which is also equal to distance of $B,D$ from respective nearest line among these. Ratio of area of smallest triangle so formed to area of parallelogram is $\frac{1}{32}$. Find $n.$