Final answer:
The lengths of the sides of triangle ABC would be AB = 3.2 inches, BC = 4.8 inches, and AC = 6.4 inches based on the given perimeter of the smaller triangle MNK. However, this does not match the given answer choices, suggesting an error in the provided options.
Step-by-step explanation:
To solve for the lengths of the sides of triangle ABC with sides in the ratio 2:3:4, we first need to acknowledge that points M, N, and K are the midpoints of the sides of triangle ABC. Thus, triangle MNK is similar to triangle ABC, and its sides are parallel to the sides of ABC and half the length.
Since the perimeter of triangle MNK equals 7.2 inches, we can set up proportions to determine the lengths of the sides of triangle ABC. Let the lengths of the sides of ABC be 2x, 3x, and 4x. The perimeter of ABC is then 2x + 3x + 4x = 9x. The perimeter of MNK is half of that, so 9x/2 = 7.2 inches. Solving for x gives us x = 7.2 * 2 / 9 = 1.6 inches.
Therefore, the lengths of the sides of triangle ABC would be: AB = 2x = 3.2 inches, BC = 3x = 4.8 inches, and AC = 4x = 6.4 inches. However, these lengths do not match any of the answer choices provided, which indicates a possible error in the given answers or in the interpretation of the question. If the provided answers were intended to represent the accurate side lengths, the proportionate lengths would need to be scaled to match one of the choices exactly.