Find the side and perimeter of a square whose diagonal is $10\mathrm{cm}$

In the given figure $\angle \mathrm{DFE}=90°$, $\mathrm{FG}\perp \mathrm{ED},\text{ If }\mathrm{GD}=8,\mathrm{FG}=12$. Find  $\mathrm{EG}, \mathrm{FD}$ and $\mathrm{EF}$.

In $△\mathrm{PQR}$, point $\mathrm{S}$ is the midpoint of side $\mathrm{QR}$. If $\mathrm{PQ}=11,\mathrm{PR}=17,\mathrm{PS}=13$, find $\mathrm{QR}$

In the given figure seg $\mathrm{PS}$ is the median of $\Delta \mathrm{PQR}$ and $\mathrm{PT}\perp \mathrm{QR}$
Prove that, ${\mathrm{PR}}^2={\mathrm{PS}}^2+\mathrm{QR}\times \mathrm{ST}+{\left(\frac{\mathrm{QR}}{2}\right)}^2$

In the given figure point $\mathrm{T}$  is in the interior of rectangle $\mathrm{PQRS}$. Prove that, ${\mathrm{TS}}^2+{\mathrm{TQ}}^2={\mathrm{TP}}^2+{\mathrm{TR}}^2$

(As shown in the figure, draw

Seg $\mathrm{AB}\Vert$ Side $SR$ and  $A-T-B$)