EXERCISE 6.4

Is it possible to have a triangle with the following sides?

$3 \mathrm{c}\mathrm{m},6 \mathrm{c}\mathrm{m},7 \mathrm{c}\mathrm{m}$

Is it possible to have a triangle with the following sides?

 $6 \mathrm{c}\mathrm{m},3 \mathrm{c}\mathrm{m},2 \mathrm{c}\mathrm{m}$

Take any point $\mathrm{O}$ in the interior of a triangle $\mathrm{P}\mathrm{Q}\mathrm{R}$. Is

$\mathrm{O}\mathrm{Q}+\mathrm{O}\mathrm{R}>\mathrm{Q}\mathrm{R}?$

Take any point $\mathrm{O}$ in the interior of a triangle $\mathrm{P}\mathrm{Q}\mathrm{R}$. Is

$\mathrm{O}\mathrm{R}+\mathrm{O}\mathrm{P}>\mathrm{R}\mathrm{P}?$

$\mathrm{A}\mathrm{M}$ is a median of a triangle $\mathrm{A}\mathrm{B}\mathrm{C}$. Is $\mathrm{A}\mathrm{B}+\mathrm{B}\mathrm{C}+\mathrm{C}\mathrm{A}>2\mathrm{A}\mathrm{M}?$ (Consider the sides of triangles $\mathrm{}\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{M}$ and $\mathrm{\Delta }\mathrm{A}\mathrm{M}\mathrm{C}.$)

$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}\mathrm{}$ is a quadrilateral. Is $\mathrm{A}\mathrm{B}\mathrm{}+\mathrm{}\mathrm{B}\mathrm{C}\mathrm{}+\mathrm{}\mathrm{C}\mathrm{D}\mathrm{}+\mathrm{}\mathrm{D}\mathrm{A}\mathrm{}>\mathrm{}\mathrm{A}\mathrm{C}\mathrm{}+\mathrm{}\mathrm{B}\mathrm{D}?$

$ABCD$ is a quadrilateral. Is $AB\mathrm{}+\mathrm{}BC\mathrm{}+\mathrm{}CD\mathrm{}+\mathrm{}DA\mathrm{}<\mathrm{}2(AC\mathrm{}+\mathrm{}BD)$?

The lengths of two sides of a triangle are $12 \mathrm{cm}$ and $15 \mathrm{cm}$.
Between what two measures should the length of the third side fall?