In $\Delta ABC$,$m \angle B = 90^\circ$. If $AC - AB = 27$ and $AC - BC = 6$,find the perimeter of $\Delta ABC$.

In $\Delta XYZ$,$\overline{XN}$ is a median. If $XY^2 + XZ^2 = 200$ and $XN = 8$,find $YZ$.

Which of the following groups correctly matches the data of Part $I$ with the data of Part $II$?

Part $I$	Part $II$
$1.$ In $\Delta ABC, AB=3, BC=4, AC=5$	$a.$ In-radius $= 1$
$2.$ In $\Delta PQR, PQ=5, QR=12, PR=13$	$b.$ In-radius $= 2$
$3.$ In $\Delta XYZ, XY=8, YZ=15, XZ=17$	$c.$ In-radius $= 3$
$4.$ In $\Delta MNP, MN=20, NP=21, MP=29$	$d.$ In-radius $= 6$

Prove that in a parallelogram,the sum of squares of all the sides is equal to the sum of squares of the diagonals.

In the figure,$O$ is a point in the interior of a triangle $ABC$,$OD \perp BC$,$OE \perp AC$,and $OF \perp AB$. Show that $AF^{2} + BD^{2} + CE^{2} = AE^{2} + CD^{2} + BF^{2}$.

In $\Delta ABC$,$m \angle A + m \angle C = m \angle B$ and $\overline{BM}$ is an altitude to $\overline{AC}$. If $AM = 16$ and $CM = 9$,find $BM$,$AB$,and $BC$.