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The Pythagorean Theorem is a key mathematical concept in geometry. It's particularly useful when dealing with right-angled triangles. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This is often written as: \(c^2 = a^2 + b^2\).

Applying this theorem helps solve various problems involving distances and lengths. For example, in the case of our square, the diagonal acts as the hypotenuse of right-angled triangles formed by dividing the square across the diagonal. Knowing this relationship makes it easier to derive formulas and solve problems related to the square's properties.

A right-angled triangle is one that contains a 90-degree angle. This type of triangle has some special properties that make calculations easier. In our exercise, when a square is split along its diagonal, two right-angled triangles are formed.

Each of these triangles shares two sides that are equal to the square's side length. The diagonal of the square becomes the hypotenuse of these triangles. By understanding the makeup of right-angled triangles, we can effectively use the Pythagorean Theorem to find missing lengths. This fundamental knowledge allows us to treat complex shapes by breaking them down into simpler right-angled triangles.

To solve for the sides of a right-angled triangle using the Pythagorean Theorem, follow these steps:

• Identify the sides: In our case, let's denote the sides of the square as 'a' and the diagonal as 'd'.
• Write the relationship: Using the Pythagorean Theorem: \(d^2 = a^2 + a^2\)
• Simplify the equation: Combine like terms to get \(d^2 = 2a^2\)
• Isolate 'a': Solve for the side length 'a' by dividing and taking the square root: \(a = \frac{d}{\text{\text{√2}}}\)

This step-by-step process shows how the properties of right-angled triangles can be used to find the side length of a square when given the diagonal. It demonstrates the power of the Pythagorean Theorem in solving practical geometric problems.