Why is the diagonal of a square shorter than the sum of the two sides?

When you stand in one corner of a square and want to reach the opposite corner, you can take two different paths.

You can move along the sides of the square by going 10 units along the x-axis and 10 units along the y-axis, covering a total distance of 20 units.

Alternatively, you can follow a zig-zag pattern, moving 1 unit along the x-axis and then 1 unit along the y-axis, and so on.

If you continue this pattern with smaller and smaller steps, it would eventually form a straight line, right?

But why is the straight diagonal line shorter than the sum of the two lengths of the square?

Understanding the Straight Line

It seems counterintuitive, but a straight line has no turns.

The line made up of zigzags won't become a straight line even if the zigzags get smaller and smaller.

A straight line is the shortest distance between two points, and it has no deviations or changes in direction.

Exploring the Pythagorean Theorem

The answer to the mystery of the shorter diagonal in a square lies in the Pythagorean theorem.

This theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

When applied to a square, the two sides are the length and width, and the hypotenuse represents the diagonal.

Application of the Theorem

In a square where each side measures 10 units, using the Pythagorean theorem, 10 squared plus 10 squared equals the length of the diagonal squared.

Simplifying, 100 plus 100 equals the diagonal squared, giving us 200.

Taking the square root of 200, we get the length of the diagonal, which is approximately 14.14 units.

Mathematical Implications

From a mathematical perspective, it's like adding two numbers and getting a third larger number, which is not intuitive.

The Pythagorean theorem provides crucial insight into understanding the relationship between the sides and the diagonal of a square, enabling us to calculate the length of the diagonal without physically measuring it.

The Realization

In essence, it's not just about the total length traveled but about how the distance is covered.

The diagonal is shorter not because the total distance is reduced, but because the nature of the diagonal as a straight line allows for a direct, uninterrupted path from one corner to the other within the square.