In a right-angled triangle, the lengths of sides containing the right angle are $20 \mathrm{cm}$ and $21 \mathrm{cm}$. WHat is the radius of the circumcircle of the trinagle?

In the given figure, $\angle \mathrm{ABC}={90}^{°}, \mathrm{AC}=13 \mathrm{cm}, \mathrm{CD}=11 \mathrm{cm}, \mathrm{AD}=20 \mathrm{cm}.$ Find BC and AB.

In the given figure, $\mathrm{PQ}$ and $\mathrm{TS}$ are perpendiculars to $\mathrm{QS}. \mathrm{R}$ is the mid-point of $\mathrm{QS}$ and $\angle \mathrm{PRT}={90}^{°}$. If $\mathrm{PQ}=9 \mathrm{cm}, \mathrm{QS}=24 \mathrm{cm}, \mathrm{TS}=16 \mathrm{cm}$, find $\mathrm{PT}$.

In $△\mathrm{ABD}$, $\mathrm{AB}\perp \mathrm{BD}$ and $\mathrm{AC}=\mathrm{BD}$. If $\mathrm{AB}= 4 \mathrm{cm}, \mathrm{BC}=2 \mathrm{cm},$ find $\mathrm{AD}$.

In $△\mathrm{ABC}, \mathrm{AB}=8$, find a and b if $\mathrm{a}+\mathrm{b}=32$.

In quadrilateral $\mathrm{ABCD}, \angle \mathrm{B}=\angle \mathrm{D}={90}^{°}.$ Prove that $2{\mathrm{AC}}^2-{\mathrm{BC}}^2={\mathrm{AB}}^2+{\mathrm{AD}}^2+{\mathrm{DC}}^2$.

In triangle $\mathrm{ABC}, \mathrm{AD}\perp \mathrm{BC}$, prove that $\left(\mathrm{AB}+\mathrm{AC}\right)\left(\mathrm{AC}-\mathrm{AB}\right)=\left(\mathrm{CD}+\mathrm{BD}\right)\left(\mathrm{CD}-\mathrm{BD}\right)$.

The diagonal of quadrilateral PQRS intersect at O, at right angles. Prove that

${\left(\mathrm{PQ}\right)}^2+{\left(\mathrm{SR}\right)}^2={\left(\mathrm{PS}\right)}^2+{\left(\mathrm{QR}\right)}^2$

In $△\mathrm{ABC}$, $\angle \mathrm{B}=90°$ and $\mathrm{D}$ is the mid point of $\mathrm{BC}$, Prove that: ${\mathrm{AC}}^2={\mathrm{AD}}^2+3{\mathrm{CD}}^2$