In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = x+3$,$AB = 2x$,$AN = x+5$ and $AC = 2x+3$,find the value of $x$.

$E$ and $F$ are points on the sides $PQ$ and $PR$ respectively of a $\Delta PQR$. For each of the following cases,state whether $EF || QR$. $PQ = 1.28 \, cm, PR = 2.56 \, cm, PE = 0.18 \, cm$ and $PF = 0.36 \, cm$.

In the figure,if $AD \perp BC$,prove that $AB^2 + CD^2 = BD^2 + AC^2$.

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM : AB = 2 : 3$ and $AC = 15$,then $NC = \ldots$

In rhombus $ABCD$,$AC = 18$ and $BD = 24$. Find the perimeter of rhombus $ABCD$.

In the figure,$ABC$ is a triangle in which $\angle ABC > 90^{\circ}$ and $AD \perp CB$ produced. Prove that $AC^{2} = AB^{2} + BC^{2} + 2BC \cdot BD$.