In a right triangle, what is the relationship between the lengths of the sides?

In a right triangle, the relationship between the side lengths is defined by the Pythagorean theorem. This theorem states that for any right triangle, the square of the length of the longest side (the hypotenuse) is equal to the sum of the squares of the lengths of the other two sides.

Mathematically, if we denote the lengths of the two shorter sides as (a) and (b), and the hypotenuse as (c), the theorem is expressed as:

[ c^2 = a^2 + b^2 ]

This fundamental relationship holds true for all right triangles, regardless of their size. It’s an essential concept in geometry that illustrates how side lengths interact in a right triangle configuration.

In contrast, the other options misrepresent the relationships in a right triangle. For example, while the statement about the ratio of 3:4:5 does describe specific integer values that form a right triangle (known as a Pythagorean triple), it does not apply to all right triangles. Similarly, the claim that the longest side is equal to the sum of the other two sides describes a property of triangles in general, not specifically of right triangles. The notion that the sum of the