Question

describe each statement as always, sometimes, or never true.
the centroid is located outside the triangle.
using the perpendicular distance between the two, the orthocenter is equidistant from each side of the triangle.
the centroid and orthocenter are located at the same point inside the triangle.
the distance from the centroid to a vertex is twice the distance from the centroid to the midpoint of the opposite side.

Explanation:

Step1: Analyze "The centroid is located outside the triangle."

The centroid of a triangle is the intersection of its medians. Medians always lie inside the triangle, so the centroid is always inside the triangle. Thus, this statement is never true.

Step2: Analyze "Using the perpendicular distance between the two, the orthocenter is equidistant from each side of the triangle."

The orthocenter is the intersection of altitudes. The incenter (not orthocenter) is equidistant from each side (as it's the center of the incircle). The orthocenter's distance to sides varies. So this is never true.

Step3: Analyze "The centroid and orthocenter are located at the same point inside the triangle."

In an equilateral triangle, centroid, orthocenter, incenter, and circumcenter coincide. In non - equilateral triangles, they are different. So this is sometimes true.

Step4: Analyze "The distance from the centroid to a vertex is twice the distance from the centroid to the midpoint of the opposite side."

This is a property of the centroid of a triangle. The centroid divides each median in a ratio of 2:1, with the longer part being from the vertex to the centroid. So this is always true.

Answer:

1. "The centroid is located outside the triangle." - Never
2. "Using the perpendicular distance between the two, the orthocenter is equidistant from each side of the triangle." - Never
3. "The centroid and orthocenter are located at the same point inside the triangle." - Sometimes
4. "The distance from the centroid to a vertex is twice the distance from the centroid to the midpoint of the opposite side." - Always