Inequalities in a Triangle

In a triangle $ABC$, let $AB= 5 \mathrm{cm}, BC= 6 \mathrm{cm} \text{and} CA= 7 \mathrm{cm}$. If $\text{'}O\text{'}$ is any point inside the triangle, then the length of $OA+OB+OC$ is:

$ABCD$ is quadrilateral. Is $AB+BC+CD+DA<2(AC+BD) ?$

Is it possible for the $8 \mathrm{ft}, 5 \mathrm{ft}, 9 \mathrm{ft}$ to be the dimensions of a triangle

Difference of two sides of a triangle is always _____ than the third side. 

The smallest side in the following triangle is $BC$

If the two angles of a triangle are unequal, then the smaller angle has the _____ side opposite to it.

Identify the largest angle of the given triangle

In the figure below, which is the largest angle and smallest angle.

In the given figure, sides $PQ$ and $PR$ of $△PQR$ are extended to points $A$ and $B$ respectively. Also, $\angle AQR<\angle BRQ$. Prove that $PR>PQ$. 

Prove that in a right-angled triangle, the longest side is hypotenuse.

In $∆ABC; \angle C$ is obtuse and $AD\perp BC$ Produced. Which of the following statement(s) are true?

$$\begin{gathered} 1. AC=c\sin B.\sin (180°-C) \\ 2. BD=a+b \cos C \\ 3. A{B}^2=B{C}^2+C{A}^2+2BC.CD \\ 4. BC=c \cos B-\left|b \cos \left(180°-C\right)\right| \end{gathered}$$

A quadrilateral has distinct integral side lengths. If the second-largest side has length $10,$ then the maximum possible length of the largest side is

In triangle $ABC, AB=\left(b^2-1\right) \mathrm{cm}, BC=a^2 \mathrm{cm}$ and $AC=2a \mathrm{cm},$ where $a$ and $b$ are positive integers greater than $1.$Find the value of $a-b.$

In a triangle with integer side lengths, one side is three times as long as a second side; and the length of the third side is $17.$ What is the greatest possible perimeter of the triangle ?

$AD$ is a median of the triangle $ABC$. If $AB+BC+CA>x$ then find $x$.

ABC is an isosceles triangle with AB = AC. D is any point online segment BC and E is a point on BC produced.
Prove that :  AE > AD 

$ABC$ is an isosceles triangle with $AB = AC.$ $D$ is any point online segment $BC$and $E$ is a point on $BC$ produced.
Prove that :  $AE > AC$

$ABC$ is an isosceles triangle with $AB = AC$.$D$ is any point on line segment $BC$ and $E$ is a point on $BC$ produced.
Prove that:  $AC > AD$

Prove that in a triangle, other than equilateral triangle, the angle opposite to the longest side is greater than $\frac{2}{3}$ of a right angle.

 In a $△$PQR, PS is drawn perpendicular to QR. If SR > QS, Show that PR > PQ.