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The Pythagorean Theorem is essential for solving problems involving right-angled triangles. It states that for any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as: \[ c^2 = a^2 + b^2 \]In a square where the diagonal splits the shape into two equal right-angled triangles, both legs of these triangles are of equal length. Therefore, if 's' is the length of one side of the square (the legs of the triangles), the formula for the diagonal (hypotenuse 'c') becomes:\[ c = s \sqrt{2} \]This means that to find the diagonal of a square, you multiply the side length by the square root of 2.

The diagonal of a square is a crucial value that can be used to find the length of the square's sides. When a square's diagonal is drawn, it forms two congruent right-angled triangles. If we know the length of the diagonal, we can use the diagonal formula derived from the Pythagorean Theorem to find the side length. Given a diagonal 'd' of 5 cm in this case, the side length 's' can be calculated using the formula:\[ s = \frac{d}{\sqrt{2}} \]Here, 'd' is the length of the diagonal, and understanding the relationship between the side and the diagonal helps in efficiently solving problems involving squares.

Rationalizing the denominator is a process used to eliminate any irrational numbers from the denominator of a fraction. This is done by multiplying both the numerator and the denominator by the conjugate of the denominator. So, if we have: \[ s = \frac{5}{\sqrt{2}} \]We multiply the numerator and the denominator by \sqrt{2} to get: \[ s = \frac{5 \sqrt{2}}{2} \]This step transforms the denominator into a rational number, making it simpler to work with in further calculations. Rationalizing the denominator ensures that our final answer is in a standard and conventional form.