Final answer:
The initial question does not provide enough information to prove that MK = PN, as there's no given relationship between these segments and the right triangles mentioned. The references to the Pythagorean theorem and right triangles are unrelated to proving MK = PN without additional context. Hence, we are unable to determine the validity of the statement with the given details.
Step-by-step explanation:
The question provided seems to contain unrelated information. The initial statement about triangles FGH and FJH being right triangles and GH = JH is not related to proving that MK = PN based on the given that L is the midpoint of both KN and MP. Instead, the proof would require additional information about the relationship between the points and lines mentioned to determine whether the segments MK and PN are of equal length.
However, the question also contains numerous references to the Pythagorean theorem, which indicates an emphasis on right triangles and computations of side lengths. The Pythagorean theorem states that for any right triangle with legs a and b, and hypotenuse c, the relationship is given by a² + b² = c². This is used to determine the length of a side if the lengths of the other two sides are known.
Without more specific information regarding the configuration of MK, PN, KN, and MP and how they relate to one another or to right triangles, the proof that MK = PN cannot be completed. Thus, the best answer to the question is Option 3: Insufficient information to determine the validity.