Answer
Since, the expression \[{{10}^{2}}\text{+ }{{11}^{2}}+{{12}^{2}}\]is written as,
\[\begin{align}
& {{10}^{2}}\text{ }+\text{ }{{11}^{2}}\text{ }+\text{ }{{12}^{2}}=10\cdot 10+11\cdot 11+12\cdot 12 \\
& =100+121+144 \\
& =365
\end{align}\] …… (1)
And, the expression \[{{13}^{2}}\text{ }+\text{ }{{14}^{2}}\]written as,
\[\begin{align}
& {{13}^{2}}\text{ }+\text{ }{{14}^{2}}=13\cdot 13+14\cdot 14 \\
& =169+196 \\
& =365
\end{align}\] …… (2)
From 1 and 2 equations, both the expression are equals also 365 is the number of days in a non-leap year.
Hence, both sides of the expression \[{{10}^{2}}\text{ }+\text{ }{{11}^{2}}\text{ }+\text{ }{{12}^{2}}\text{ }=\text{ }{{13}^{2}}\text{ }+\text{ }{{14}^{2}}\]are equal and the significance of the sum is that sum is the number of days in a non-leap year.