Which of the following represents the Pythagorean theorem?

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle, denoted as c) is equal to the sum of the squares of the lengths of the other two sides (denoted as a and b). This relationship is precisely captured by the formula a² + b² = c².

Understanding this relationship is crucial in various applications of mathematics and physics, as it allows for the calculation of distances, angles, and other geometric properties. The correctness of this theorem is established by the nature of right triangles and has been proven through various mathematical methods.

The other choices do not accurately represent the Pythagorean theorem. For instance, c² = a² - b² does not hold true for right triangles, as it incorrectly suggests a subtraction rather than addition. The options a + b = c and c = a + b imply linear relationships rather than the quadratic relationship involved in the theorem, making them unsuitable representations of the Pythagorean theorem.