\( \left. \begin{array} { l } { a ^ { 2 } = b ^ { 2 } + c ^ { 2 } } \\ { a ^ { 2 } = 10 ^ { 2 } - 5 ^ { 2 } } \\ { a ^ { 2 } = 100 - 25 } \\ { a ^ { 2 } = 75 } \end{array} \right. \)

Did you know that the equation \(a^2 = b^2 + c^2\) is a representation of the Pythagorean theorem, which dates back to ancient Greece? This theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle, represented by \(a\)) is equal to the sum of the squares of the other two sides (represented by \(b\) and \(c\)). Mathematicians like Pythagoras contributed immensely to our understanding of geometry! In the problem, you found \(a^2 = 75\), which means \(a\) can be calculated by taking the square root of \(75\). However, a common mistake here is neglecting to simplify \(\sqrt{75}\). It can be expressed as \(5\sqrt{3}\), so don’t forget to present your answer in its simplest form! Remember, when squaring numbers, it’s easy to overlook simplification steps!