In triangle ABC, D is the centroid and M is the midpoint of segment AC. If BD=x+9 and DM=6, find MB. Answer choices: 9,18, or 3.

To find the length of MB, we need to use the properties of a centroid and a midpoint in a triangle.

The centroid of a triangle divides each median in the ratio 2:1. This means that the length of BD is twice the length of DM. Given that DM = 6, we can compute the length of BD by doubling it (2 * DM = BD).

BD = 2 * DM = 2 * 6 = 12.

Now, since M is the midpoint of segment AC, we know that AM = MC. Therefore, the length of AM is equal to half the length of AC.

In a triangle, the centroid divides each median into two segments such that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint. In other words, the length of DM is half the length of MC. Therefore, MC = 2 * DM = 2 * 6 = 12.

Now, we have the lengths of BD and MC. To find the length of MB, we can subtract the length of BD from the length of MC.

MB = MC - BD = 12 - 12 = 0.

Therefore, the length of MB is 0. So, the correct answer choice is 0, which is not listed in the answer choices provided.