Embibe Experts Solutions for Chapter: Properties of Triangle, Exercise 1: Level 1

Let $ABC$ be an acute angled triangle with circumcentre $O$ and orthocenter $H.$ If $AO=AH,$ then the measure of angle $A$ is:

If in a $∆ABC, 3a=b+c,$ then the value of  $\cot \frac{B}{2}·\cot \frac{C}{2}$ is

In a $\Delta ABC,\angle B=\frac{\pi }{2},$ then value of radius of incircle is:

In a triangle $ABC$, if $\cot \frac{A}{2}\cot \frac{B}{2}=c,\cot \frac{B}{2}\cot \frac{C}{2}=a$ and $\cot \frac{C}{2}\cot \frac{A}{2}=b$, then $\frac{1}{s-a}+\frac{1}{s-b}+\frac{1}{s-c}=$

If in a triangle $ABC, r_1+r_2+r_3=9r,$ then the triangle is necessarily

If $a,b$ and $c$ are the sides of a triangle such that $a^4+b^4+c^4=2c^2\left(a^2+b^2\right)$, then the angles opposite to the side $c$ is

The diagram shows an equilateral triangle $ADE$ inside a square $ABCD.$ What is the value of ${\left(\frac{ \mathrm{area} \mathrm{of} \Delta ADE}{ \mathrm{area} \mathrm{of} \Delta DEC}\right)}^2$?

A triangle $∆ABC$ is inscribed in a circle of radius $1,$ with $\angle BAC=60°.$ Altitudes $AD$ and $BE$ of $∆ABC$ intersect at $H.$ Find the smallest possible value of the length of the segment $AH\mathit{.}$