If $\Delta ABC \sim \Delta DEF$ for the correspondence $ABC \leftrightarrow DEF$,and given $AB = 8$,$AC = 10$,$DE = 12$,and $EF = 18$,find the perimeter of $\Delta DEF$.

The two triangles in the figure are congruent using a congruence theorem. It is given that $OQ = OR$. Which of these conditions,along with the given condition,is sufficient to prove that the two triangles are congruent to each other?

For $\Delta ABC$ and $\Delta XYZ$,$\frac{AB}{XZ} = \frac{BC}{XY} = \frac{AC}{YZ}$. Then,the correspondence $ABC \leftrightarrow \ldots \ldots$ between them is a similarity.

For going to a city $B$ from city $A$,there is a route via city $C$ such that $AC \perp CB$,$AC = 2x \, km$ and $CB = 2(x+7) \, km$. It is proposed to construct a $26 \, km$ highway which directly connects the two cities $A$ and $B$. Find how much distance will be saved in reaching city $B$ from city $A$ after the construction of the highway (in $km$).

In $\square PQRS$,$\overline{PQ} \parallel \overline{RS}$ and $\overline{PR} \cap \overline{QS} = \{X\}$. If $XP = 2x + 4$,$XQ = 4x - 2$,$XR = x + 1$,and $XS = 4$,find the value of $x$.

If $\triangle ABC \sim \triangle DEF$,$AB = 4 \, cm$,$DE = 6 \, cm$,$EF = 9 \, cm$,and $FD = 12 \, cm$,find the perimeter of $\triangle ABC$ (in $cm$).