Final answer:
Based on the properties of medians in a triangle, we can derive lengths of various segments by following the rule that a median divides a side into two equal parts and at the centroid into a 2:1 ratio. The lengths are OT = 6, BO = 12, IT = 6, TR = 12, TS = 18, and US = 27.
Step-by-step explanation:
In the context of a triangle with medians, we can derive the lengths by understanding that the medians of a triangle divide each other into segments with a specific ratio. In a triangle, each median divides the opposite side into two equal parts, and it is also divided into two parts by the centroid (where all medians intersect), with the piece from the vertex to the centroid being twice as long as the piece from the centroid to the middle of the side.
Let's go through the provided measurements step by step:
- OT: Since BT is 12 and OT is half of it (as a median), OT = 12 / 2 = 6.
- BO: BO is the full length of the median, i.e., twice of OT, so BO = 6 * 2 = 12.
- IT: IR is a median and is 18 long. The median is divided in a 2:1 ratio by the centroid, thus IT being 1/3 of IR, IT = 18 / 3 = 6.
- TR: Being the other segment of the median IR, TR = 2 * IT = 2 * 6 = 12.
- TS: UT is the shortest piece which should be 1/3 of US since UT is the segment towards the vertex of the median US. So, if UT is 9 long, US is thrice that, US = 9 * 3 = 27. Therefore, TS being twice of UT will be 9 * 2 = 18.
- BR: Given BU is (4x - 6) and RU is (2x + 12), for them to meet at U and form a median, their lengths must sum to the length of BR. Thus, BR = BU + RU = (4x - 6) + (2x + 12). By substituting x so they sum up to the known lengths, we can set x accordingly and find BR's length, which in this case is not feasible since we don't have a specific length for BU or RU.
Considering the possible answers and the lengths we've calculated, option b seems correct: OT = 6, BO = 12, IT = 6, TR = 12 (not 18, which seems to be a typo in the options provided), TS = 18, US = 27.