To get from point $\mathrm{A}$ to point $\mathrm{B}$ you must avoid walking through a pond. You must walk $34 \mathrm{m}$ south and $41 \mathrm{m}$ east. To the nearest meter, how many meters would be saved if it were possible to make a way through the pond?(Write the answer in $\mathrm{m}$).

In the rectangle $\mathrm{WXYZ}, \mathrm{XY}+\mathrm{YZ}=17 \mathrm{cm},$and $\mathrm{XZ}+\mathrm{YW}=26 \mathrm{cm}.$ Calculate the length and breadth of the rectangle?

The hypotenuse of a right triangle is $6 \mathrm{m}$ more than twice of the shortest side. If the third side is $2 \mathrm{m}$ less than the hypotenuse, find the sides of the triangle?

The perpendicular $\mathrm{PS}$ on the base $\mathrm{QR}$ of $∆\mathrm{PQR}$ intersects $\mathrm{QR}$ at $\mathrm{S},$ such that $\mathrm{QS}=3 \mathrm{SR}.$ Prove that $2{\mathrm{PQ}}^2=2{\mathrm{PR}}^2+{\mathrm{QR}}^2.$

In the adjacent figure, $\mathrm{ABC}$ is a right-angled triangle with right angle at $\mathrm{B}$ and points $\mathrm{D},$ $\mathrm{E}$ trisect $\mathrm{BC}$. Prove that $8{\mathrm{AE}}^2=3{\mathrm{AC}}^2+5{\mathrm{AD}}^2.$