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

In the figure given below, $ABCD$ is a quadrilateral in which $AB= AD$,

$\angle A = 90˚ = \angle C, BC = 8 cm and CD = 6 cm.$ Find $AB$ and calculate the area of $∆ABD$.

In the above figure, $\angle B=90°$ and $\angle D=90°$. If $AB=12 \mathrm{cm}, AC=13 \mathrm{cm}, CE=10 \mathrm{cm}$ and $ED=6 \mathrm{cm}$, find the length of $BD$.

In the figure given below, $\angle PSR = 90°, PQ= 10 \mathrm{cm}, QS=6 \mathrm{cm} \mathrm{and} RQ=9 \mathrm{cm}$. Calculate the length of $PR$ (in $\mathrm{cm}$).

In the adjoining figure, $\angle D=90°$. If $AB=16 \mathrm{cm}, BC=12 \mathrm{cm}$ and $CA=6 \mathrm{cm}$, find the length of $CD.$

In the figure given below$, BC=5 \mathrm{cm}, \angle B=90°, AB=5AE, CD=2AE \mathrm{and} AC=ED.$ Calculate the lengths of $EA, CD, AB \mathrm{and} AC.$

In the adjoining figure, $ABC$ is a right-angled triangle at $C$. If $D$ is midpoint of $BC$, prove that $A{B}^2=4A{D}^2-3A{C}^2$.