Final answer:
The given information about the lengths AD=8 and AG=10 seems inconsistent when considering the centroid division rule, which suggests an error in the provided lengths. Proper calculation of segments requires reevaluating the given measures or assuming a typo and correcting it to maintain the centroid's median division ratio.
Step-by-step explanation:
The question pertains to finding the lengths of segments in a triangle where the centroid is given. The centroid of a triangle, denoted as G, divides each median into segments with a ratio of 2:1, with the longer segment being closer to the vertex opposite the side of the median. Given AD=8, AG=10, and CD=18, we can use these properties to find the lengths.
Since AG is two-thirds the length of the median AD (because G is the centroid), the length of GD (the remaining third of the median) is half the length of AG, which is 5. Therefore, we have the lengths of AG = 10, and GD = 5, and since AG + GD = AD, this confirms that AD indeed equals 8 (10 + 5 = 15, which is not given information, hence GD is incorrect and we should reevaluate our approach).
To correct, as AG is 10 and AD is 8, GD must be AD - AG, which equals 8 - 10 = -2. The negative result suggests a mistake since lengths cannot be negative. Reassessing, DG actually should be one-third of AD, as AG would be two-thirds of the entire median, not the other segment. Therefore, AG at two-thirds would be 8 * (2/3) = 16/3 and GD as one-third would be 8/3. The question might contain a typo, as AG cannot be larger than the whole median AD. If AG is indeed 10, then AD must be wrongly stated which needs correction.
For CD, if we have CD = 18, CG would be twice that of GD, but without the correct length of GD, we cannot determine CG accurately. Assuming AD is actually 15 (given AG is two-thirds), GD would be 5, making CD = 18 correct, and CG then is 2 * 5 = 10.