Final answer:
Katherine's solution involves the Pythagorean Theorem which is used to calculate the hypotenuse of a right triangle and is essential in the analytical method for finding vector magnitudes and directions.
Step-by-step explanation:
Katherine's solution to the problem with the re-entry of the capsule involves the Pythagorean Theorem, which relates the lengths of the legs of a right triangle to its hypotenuse. This relationship is expressed as a² + b² = c², which can be solved for c giving c = √a² + b². This theorem is fundamental in the analytical method used to determine the magnitude and direction of resultant vectors, as well as solving problems related to right-angled triangles, such as those that might be encountered in engineering or physics scenarios, like the re-entry of space capsules.