Embibe Experts Solutions for Chapter: Properties of Triangle, Exercise 1: IOQM - PRMO and RMO 2020

The sides of a triangle $arcx,2x+1$ and $x+2$ for some positive rational number $x.$ If one angle of the triangle is $60°,$ what is the perimeter of the triangle?

Let $ABC$ be a triangle with $\angle BAC=90°$ and $D$ be the point on the side $BC$ such that $AD\perp BC.$ Let $r, r_1$ and $r_2$ be the inradius of triangles $ABC, ABD,$ and $ACD$, respectively. If $r, r_1,$ and $r_2$ are positive integers and one of them is $5$ find the largest possible value of $r+r_1+r_2$.

Let $ABCD$ be a convex cyclic quadrilateral. Suppose $P$ is a point in the plane of the quadrilateral such that sum of its distances from the vertices of $ABCD$ is the least. If $\{PA,PB,PC,PD\}=\left\{3,4,6,8\right\}$, what is the maximum possible area of $ABCD?$

If $15$ and $9$ are lengths of two medians of a triangle, what is the maximum possible area of the triangle to the nearest integer?

Let $ABC$ be an isosceles triangle with $AB=BC\mathit{.}$ A trisector of $\angle B$ meets $AC$ at $D\mathit{.}$ If $AB\mathit{,}\mathit{ }AC$ and $BD$ are integers and $AB-BD=3,$ find $AC$.

A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$. An ant at the mid-point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass (See the figure.). If $d$ the shortest distance it should crawl to reach the honey drop, what is the integer part of $d?$ (ignore the thickness of the glass.)

In a triangle $ABC\mathit{,}$ it is known that $\angle A=100°$ and $AB=AC\mathit{.}$ The internal angle bisector $BD$ has length $20$ units. Find the length of $BC$ to the nearest integer, given that $\sin 10°\approx 0.174.$

In a triangle $ABC,$ right-angled at $A,$ the altitude through $A$ and the internal bisector of $\angle A$ have lengths $3$ units and $4$ units, respectively. Find the length of the medium through $A.$