Final answer:
L' is located at (0,3) after a 90-degree rotation of triangle KLM about point K, making option c) correct. Additionally, K'L' is the hypotenuse of the newly formed right triangle K'L'M', confirming option b). Both option a) and d) are incorrect.
Step-by-step explanation:
When the triangle KLM is rotated 90 degrees about point K, the image of point L, which is initially at (3,0), will move to a new position L'. Because L is 3 units away from K along the x-axis, after a 90-degree rotation, L' will be 3 units away from K along the y-axis, so L' will be at (0,3). Therefore, option c) L' is located at (0,3) is true.
The line segment K'L' will not be parallel to KM because KM is not horizontal or vertical; instead, KM has a positive slope, and after rotation, K'L' will have a negative slope. Therefore, option a) is incorrect. Option b) is correct because K'L' becomes the hypotenuse of right triangle K'L'M', with KL' and KM' being the perpendicular sides after the rotation. Lastly, option d) is incorrect because, whereas KL is horizontal before rotation, L'M' (formerly KM) will be vertical after rotation and therefore is not parallel to KL.