In $\Delta ABC$ and $\Delta PQR$,$\angle A \cong \angle P$ and $\angle B \cong \angle R$. Then,the correspondence $ABC \leftrightarrow \ldots$ is a similarity.

It is given that $\triangle ABC \sim \triangle PQR,$ with $\frac{BC}{QR} = \frac{1}{3}.$ Then,$\frac{\operatorname{ar}(\triangle PRQ)}{\operatorname{ar}(\triangle BCA)}$ is equal to

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 $\Delta ABC$,$\overline{AD}$ is a median. If $AB^2 + AC^2 = 338$ and $AD = 5$,find $BC$.

In $\Delta ABC$,$P$,$Q$,and $R$ are the midpoints of $\overline{AB}$,$\overline{BC}$,and $\overline{CA}$ respectively. Then,which of the following statements is not true?

In square $ABCD$,$AB = 4 \sqrt{2}$. Then,the length of a diagonal of the square is......