The areas of two similar triangles are $200$ and $128$. Then,the ratio of their corresponding altitudes is.....

$A$ vertical pole of length $6\, m$ casts a shadow $4\, m$ long on the ground and at the same time a tower casts a shadow $28\, m$ long. Find the height of the tower. (in $m$)

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $\overline{AC}$. Then,the correspondence $ABC \leftrightarrow \ldots$ between $\Delta ABC$ and $\Delta ADB$ is a similarity.

If $\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$,and $\frac{AB}{4} = \frac{XY}{5}$,then $\frac{BC}{YZ} = \ldots$

If $\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow ZYX$,and given $PQ = 6$,$QR = 8$,and $XY = 12$,find the length of $YZ$.

In the figure,two chords $AB$ and $CD$ intersect each other at the point $P$. Prove that $AP \cdot PB = CP \cdot DP$.