Final answer:
The length of the sides of triangle ABC, when given the ratio 2:3:4 and that the perimeter of the inscribed triangle MNK is 7.2 inches, are calculated to be 3.2 inches, 4.8 inches, and 6.4 inches.
Step-by-step explanation:
When we are given that the ratio of the lengths of the sides of triangle ABC is 2:3:4 and that points M, N, and K are the midpoints of the sides, this leads us to consider triangle MNK, which is similar to triangle ABC but at half the scale. Thus, each side of triangle MNK is half the length of the corresponding side in triangle ABC. The perimeter of triangle MNK is given as 7.2 inches.
Let the sides of triangle ABC be represented as 2x, 3x, and 4x. Since triangle MNK is half the size of triangle ABC, the sides of triangle MNK will be x, 1.5x, and 2x. The perimeter of triangle MNK is the sum of its sides: x + 1.5x + 2x = 4.5x. Since we know that 4.5x equals 7.2 inches, we can solve for x: 4.5x = 7.2, so x = 7.2 / 4.5, which equals 1.6 inches.
Therefore, the sides of triangle ABC are:
- 2x = 2(1.6) = 3.2 inches
- 3x = 3(1.6) = 4.8 inches
- 4x = 4(1.6) = 6.4 inches