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Medians in Triangles

Overview

A median of a triangle runs from a vertex to the midpoint of the opposite side. Every triangle has three medians, and they meet at the centroid — the center of gravity of the triangle. The centroid has a measuring property that drives most of this lesson: it sits two-thirds of the way along each median from the vertex, so the vertex-to-centroid piece is twice as long as the centroid-to-midpoint piece.

Vocabulary

Median of a Triangle
A segment that goes from a vertex of a triangle to the midpoint of the opposite side.
Midpoint
The point on a segment that divides it into two congruent segments.
Concurrent
Three or more lines or segments that intersect at a single point.
Centroid
The point where the three medians of a triangle meet; it is the center of gravity of the triangle and lies two-thirds of the distance from each vertex to the midpoint of the opposite side.

Main Ideas

  1. A median connects a vertex to the midpoint of the opposite side, so it splits that side into two equal lengths.
  2. A triangle has up to three medians, and they are concurrent at the centroid.
  3. The centroid divides each median in a 2 to 1 ratio: the piece from the vertex to the centroid is twice the piece from the centroid to the midpoint, and the vertex piece is two-thirds of the whole median.
  4. The midpoint of a side can be found from coordinates by averaging the endpoints, which locates the far end of a median on the coordinate plane.

Lesson Notes

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Notes page 1

Lesson Video

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Practice Problems

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Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.