The perimeter of rhombus $ABCD$ is $116$. If $AC = 42$, find $BD$.

The perimeter of rectangle $ABCD$ is $28$. If $AC = 10$ and $AB < BC$,find the measures of its sides.

For the correspondence $ABC \leftrightarrow XYZ$ being a similarity, which of the following correctly matches the information in Part $I$ and Part $II$?

Part $I$	Part $II$
$1.$ In $\Delta ABC$ and $\Delta XYZ, \frac{AB}{XY} = \frac{BC}{YZ}$ and $\angle B \cong \angle Y$	$a. SSS$ condition
$2.$ In $\Delta ABC$ and $\Delta XYZ, \angle A \cong \angle X, \angle B \cong \angle Y$ and $\angle C \cong \angle Z$	$b. SAS$ condition
$3.$ In $\Delta ABC$ and $\Delta XYZ, \frac{AB}{XY} = \frac{BC}{YZ} = \frac{CA}{ZX}$	$c. AAA$ condition
-	$d. \text{None of the conditions apply}$

$A$ triangle whose sides measure ............ is a right-angled triangle.

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $\overline{AC}$. If $BD = 2CD$,prove that $AC = 5CD$.

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