Final answer:
To find the lengths of the sides of triangle ABC, use the ratios given and solve an equation involving the perimeters of the smaller triangle. The lengths of the sides are approximately 3.0 inches, 4.5 inches, and 6.0 inches.
Step-by-step explanation:
To find the lengths of the sides of triangle ABC, we need to determine the values of x, which represents the length of KD, and R, which represents the length of AD.
Since M, N, and K are the midpoints of the sides, we can calculate the length of AM as 1/2 of AB, AN as 1/2 of AC, and AK as 1/2 of BC.
Using the ratios of the lengths of the sides (2:3:4), we can set up the equation AM + AN + AK + NK = 7.2. Substituting the values in terms of AB and AC, we get (1/2)(AB) + (1/2)(AC) + (1/2)(BC) + (1/2)(AB/2) = 7.2.
Simplifying the equation, we find that AB + AC + BC = 7.2 * 2 = 14.4.
Since AB is 3x and AC is 4x according to the ratios, we can substitute these expressions into the equation to find 7x + 4x + 3x = 14.4. Solving for x, we get x = 14.4 / 14 = 1.0286.
Therefore, the length of AB is 3 * 1.0286 ≈ 3.086 inches, the length of AC is 4 * 1.0286 ≈ 4.115 inches, and the length of BC is 7.2 - 3.086 - 4.115 ≈ 0.999 inches.
Hence, the correct answer is B) 3.0 inches, 4.5 inches, 6.0 inches.