In figure, the angle $BAC$ is a right angle and $AD$ is perpendicular to $BC$ ; $AB=4 \mathrm{cm}, AC=3 \mathrm{cm}$ and$BD=x \mathrm{cm}$. Calculate $x$.

In the figure, the angle $BAD$ and $ADC$ are right angles and $AX$ is parallel to $BC$. If $AB=BC=5 \mathrm{cm}$, and $DC=8 \mathrm{cm}$, then the area of $ABCX$ is $k {\mathrm{cm}}^2$. Find the value of $k$.

In the figure, $AB=BC=CA=2a$ and segment $\mathrm{AD}\perp \mathrm{side} \mathrm{BC}$. show that area of $△ABC=a^2\sqrt{3}$.

In the figure, prove that $A{B}^2-A{D}^2=C{D}^2-C{B}^2$.

In a $△ABC, AD\perp BC$. Prove that: ${\mathrm{AB}}^2+{\mathrm{CD}}^2={\mathrm{AC}}^2+{\mathrm{BD}}^2$.

In a quadrilateral $ABCD$, the diagonal $AC, BD$ intersect at right angles. Prove that $A{B}^2+C{D}^2=A{C}^2+B{D}^2$.