In rectangle $ABCD$,$AB + BC = 23$ and its area is $120$. If $AB > BC$,then $AB = \ldots$

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

The lengths of the sides of a triangle are $m^{2}+n^{2}$,$2mn$,and $m^{2}-n^{2}$,where $m > n > 0$. Prove that the triangle is a right-angled triangle.

The length of a diagonal of a square is $12$. Then,the length of each side of the square is...........

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. If $AB = 7$ and $BC = 24$,find the length of $BM$.

Point $O$ lies in the interior of $\Delta PQR$. Given $O-A-P$,$O-B-Q$,$O-C-R$,$\overline{AB} \parallel \overline{PQ}$ and $\overline{BC} \parallel \overline{QR}$. Prove that $\overline{AC} \parallel \overline{PR}$.