$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. If $AB = 12, XY = 8$ and the area of $\Delta ABC$ is $81$,find the area of $\Delta XYZ$.

In $\Delta ABC$,$m \angle B = 45^{\circ}$ and $\overline{AM}$ is an altitude,$M \in \overline{BC}$. If $BC = 7$ and $AM = 4$,then $AC = \ldots$

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?

In parallelogram $ABCD$,$P$ is a point on side $BC$. The line segment $DP$ intersects the extension of side $AB$ at point $L$. Prove that: $(1) \frac{DP}{PL} = \frac{DC}{BL}$ and $(2) \frac{DL}{DP} = \frac{AL}{DC}$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $AC$. If $AD = 9$ and $CD = 4$,find $BD$.

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