Do some reading or research on the orthocenter of a...

Do some reading or research on the orthocenter of a triangle, and the centroid of a triangle. How are they found? What are their properties? Use the ideas and skills from this section to find the (a) orthocenter and (b) centroid of the triangle formed by the points $A(-8,2), B(-2,-6),$ and $C(4,0).$

Key Concepts

Orthocenter

The orthocenter is the point where all three altitudes of a triangle intersect. An altitude of a triangle is a perpendicular drawn from a vertex to the line containing the opposite side. Its location can vary depending on the type of triangle (inside for acute triangles, on the vertex for right triangles, and outside for obtuse triangles). Finding the orthocenter typically involves determining the equations of two altitudes and solving for their intersection point.

Centroid

The centroid is the point where all three medians of a triangle intersect. A median is a line segment connecting a vertex of the triangle with the midpoint of the opposite side. The centroid is often considered the center of mass or balance point of the triangle and has the unique property of dividing each median into a 2:1 ratio, with the longer segment being closer to the vertex. It is usually computed by averaging the coordinates of the vertices.

Altitudes

Altitudes are key to finding the orthocenter. An altitude is drawn from a vertex and is perpendicular to the line containing the opposite side. In coordinate geometry, calculating an altitude involves determining the slope of the side, finding the negative reciprocal of that slope for the perpendicular, and then creating an equation of the line passing through the vertex with that perpendicular slope.

Medians

Medians are fundamental in determining the centroid of a triangle. Each median connects a vertex of the triangle with the midpoint of the opposing side. The midpoint is found by averaging the coordinates of the endpoints of the side. Since the centroid is the intersection of the medians, its coordinates can be calculated as the average of the coordinates of the triangle's vertices.

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