Final answer:
The distance from the driveway, point E, to the gazebo, point G, which is the centroid of a triangular backyard, is 30 feet. This is calculated using the property that the centroid divides each median into a 2:1 ratio with the longer segment adjoining the vertex.
Step-by-step explanation:
The student is asking about the distance from a point on the median of a triangle to the centroid of the triangle. The centroid is the point where the three medians of the triangle intersect, and it has a special property in that it divides each median into segments that are in the ratio 2:1, with the larger segment being closer to the vertex of the triangle. Since the given median BE¯¯¯¯¯ is 90 feet long, the distance from the vertex B to the centroid G is 2/3 of 90 feet, and the distance from the centroid G to the mid-point E (which is where the driveway is) is 1/3 of 90 feet.
Therefore, to find the distance from the driveway, point E, to the gazebo, point G, we calculate 1/3 of 90 feet, which equals 30 feet. So, the distance from the driveway to the gazebo, which is at the centroid of the triangular backyard, is 30 feet. This application of the centroid's properties in geometry helps accurately determine the positioning of structures within a given area.