Final answer:
Upon applying the Pythagorean Theorem, statements B (27 and 45 yielding the other side as 36) and D (32 and 40 yielding the other side as 24) are correct. Other statements A, C, and E are incorrect due to the incorrect calculation of the second leg of the triangle.
Step-by-step explanation:
To apply the Pythagorean Theorem which states a² + b² = c², let's evaluate each of the student's statements to see if they make correct statements about the lengths of the sides in a right triangle:
A) If one side is 9 and the hypotenuse is 15, then the other side should be calculated as ∙(15² - 9²) = ∙(225 - 81) = ∙(144) = 12. Therefore, statement A is incorrect because the other side is 12, not 13.
B) If one side is 27 and the hypotenuse is 45, then the other side should be calculated as ∙(45² - 27²) = ∙(2025 - 729) = ∙(1296) = 36. Statement B is correct.
C) If one side is 14 and the hypotenuse is 50, then the other side should be ∙(50² - 14²) = ∙(2500 - 196) = ∙(2304) = 48. Therefore, statement C is incorrect because the other side is 48, not 46.
D) If one side is 32 and the hypotenuse is 40, then the other side should be ∙(40² - 32²) = ∙(1600 - 1024) = ∙(576) = 24. Statement D is correct.
E) If one side is 30 and the hypotenuse is 34, then the other side should be ∙(34² - 30²) = ∙(1156 - 900) = ∙(256) = 16. Therefore, statement E is incorrect because the other side is 16, not 30.
The correct statements applying the Pythagorean Theorem are B and D.