In $\Delta ABC, m\angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = x - 1$,$BM = x + 1$,and $CM = x + 4$,find the value of $x$.

$ABCD$ is a trapezium in which $AB \parallel DC$ and $P$ and $Q$ are points on $AD$ and $BC$ respectively such that $PQ \parallel DC$. If $PD = 18 \, cm$,$BQ = 35 \, cm$ and $QC = 15 \, cm$,find $AD$ (in $cm$).

In $\Delta ABC,$ the bisector of $\angle A$ intersects $\overline{BC}$ at $D.$ If $AB = 6,$ $BD = 4$ and $DC = 3,$ find $AC$.

Sides of a triangle are given below. Determine whether it is a right triangle. In case of a right triangle,write the length of its hypotenuse.
$3 \text{ cm}, 8 \text{ cm}, 6 \text{ cm}$

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BM}$ is a median. If $AC = 20$,then $BM = \ldots$

Prove that the sum of the squares of the sides of a rectangle is equal to the sum of the squares of its diagonals.