In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QD}$ is an altitude. If $PD = 9 DR$,then $PQ = \ldots \ldots \ldots \times QR$.

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QD}$ is an altitude to the hypotenuse $\overline{PR}$. If $PQ = 4QR$,prove that $PD = 16RD$.

$PQR$ is a triangle right-angled at $P$ and $M$ is a point on $QR$ such that $PM \perp QR$. Show that $PM^{2} = QM \cdot MR$.

In $\Delta ABC$,$\Delta PQR$,and $\Delta XYZ$,the correspondences $ABC \leftrightarrow QPR$ and $PQR \leftrightarrow YZX$ are similarities. Then,the correspondence $ABC \leftrightarrow \dots$ is a similarity.

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QD}$ is an altitude to the hypotenuse $\overline{PR}$. If $PQ = 3QR$,prove that $PD = 9RD$.

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 $x$.