The time period of simple pendulum is ‘T’. When the length increases by 10 cm, its period is${\text{T}}_{\text{1}}$ . When the length is decreased by 10 cm, its period is ${\text{T}}_2$ . Then the relation between $\text{T},{\text{T}}_{\text{1}}{\text{and T}}_2$  is

Time period of simple pendulum $=\text{T}$  at length$l$  .

At length$l+10$  , time period$={\text{T}}_1$

At length$l-10$  , time period$={\text{T}}_{\text{2}}$

$\text{T}=2\times \sqrt{\frac{l}{\text{g}}}\to \left(i\right)$

${\text{T}}_1=2\times \sqrt{\frac{l+10}{\text{g}}}\to \left(ii\right)$

${\text{T}}_2=2\times \sqrt{\frac{l-10}{\text{g}}}\to \left(iii\right)$

$\frac{{\text{T}}_1}{{\text{T}}_2}=\frac{2\times \sqrt{\frac{l+10}{\text{g}}}}{2\times \sqrt{\frac{l-10}{\text{g}}}}=\frac{\sqrt{l+10}}{\sqrt{l-10}}$

$\frac{{\text{T}}_1^2}{{\text{T}}_2^2}=\frac{l+10}{l-10}$

${\text{T}}_1^2=\frac{4{\pi }^2}{\text{g}}\left(l+10\right)\to \left(iv\right)$

${\text{T}}_2^2=\frac{4{\pi }^2}{\text{g}}\left(l-10\right)\to \left(v\right)$

${\text{T}}^{\text{2}}=\frac{4{\pi }^2}{\text{g}}\left(l\right)\to \left(vi\right)$

$\text{from }\left(iv\right),\left(v\right)\&\left(vi\right)$

${\text{T}}_2^2+{\text{T}}_1^2=2{\text{T}}^{\text{2}}$