Question

In the triangle ATUV, the medians are represented by TX, UY, and VW, and they intersect at a single point Z, which is the centroid of the triangle. Given that UZ = 14, VW = 66, and ZX = 12, please find the following lengths. Note that the figure is not drawn to scale.

Answer 1

Using the properties of the centroid, the lengths of the medians of triangle ATUV and segments within those medians were found.

Step-by-step explanation:

The student's question is concerned with finding specific lengths in a triangle given certain properties. We know that the centroid (Z) of a triangle divides each median into two segments in the ratio of 2:1, with the centroid being twice as close to the midpoint of the side of the triangle as it is to the vertex. Since UZ = 14, then segment VZ, which is part of the median VW, must be twice as long, meaning that VZ = 28.

The full median VW is the sum of VZ and UZ, which is 28 + 14 = 42. For median TX where ZX is given as 12, we use the same 2:1 ratio to find XT. Since ZX is the shorter segment (from the centroid to the midpoint of the side), XT must be twice as long, so XT = 24.

To find VW, we need to realize that VW is the entire median, and we have already determined that UZ is 14 and VZ is 28, so VW = VZ + UZ = 28 + 14 = 42.