Given $\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow XYZ$. If $\frac{PQ}{3} = \frac{XY}{5}$ and $QR = 9$,find the length of $YZ$.

In $\Delta ABC$ and $\Delta DEF$,$\angle A \cong \angle E$ and $m \angle A + m \angle B = m \angle D + m \angle E$. Then the correspondence $ABC \leftrightarrow \ldots$ between them is a similarity.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. Then,$AB$ is the geometric mean of.......

The areas of two similar triangles are $144$ and $81$. If the length of the longest side of the first triangle is $36$,the length of the longest side of the second triangle is........

$\Delta ABC$ and $\Delta XYZ$ are similar under the correspondence $ABC \leftrightarrow XZY$. If $AB = 6$,$XY = 12$,$YZ = 6$,and $ZX = 9$,find the perimeter of $\Delta ABC$.

In the figure,the line segment $XY$ is parallel to side $AC$ of $\Delta ABC$ and it divides the triangle into two parts of equal areas. Find the ratio $\frac{AX}{AB}$.