In the given figure,$\angle BAC = 90^{\circ}$ and $AD \perp BC$. Then,

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QD}$ is an altitude to the hypotenuse $PR$. If $QD = 15$ and $PR = 34$,find $PQ$.

In a quadrilateral $ABCD$,$\angle A + \angle D = 90^{\circ}$. Prove that $AC^{2} + BD^{2} = AD^{2} + BC^{2}$.

$\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow XYZ$. If $PQ : QR : PR = 3 : 5 : 7$ and the perimeter of $\Delta XYZ$ is $22.5$,then $YZ = \ldots$

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 12$ and $CM = 3$,then $BM = \dots$

Corresponding sides of two similar triangles are in the ratio of $2:3$. If the area of the smaller triangle is $48 \, cm^2$,find the area of the larger triangle (in $cm^2$).