Geometry: Medians and Centroids of Triangles

Median of a Triangle

A median of a triangle connects a vertex to the midpoint of the opposite side.

Points and Relationships

• In triangle AYWT:
  • Medians: WG, TD, YF
  • Midpoints:
  • P is midpoint of WH
  • R is midpoint of TH
  • K is midpoint of YH

Length Relationships

Conjecture about the Centroid

• The centroid (intersection of medians) is twice the distance from a vertex compared to the distance to the midpoint of the opposite side.

Triangle Midsegment Theorem

• The midsegment connects midpoints of two sides, is half the length of the third side, and parallel to it.

Proof Concept

Given triangles ASGC and AHTC with midpoints defined, it states that if SH = HC and GT = TC, then SG is parallel to HT based on the midsegment theorem.

Finding the Centroid Coordinates

For triangle AMBG with vertices M(0, -3), B(8, 0), G(13, -7):

1. Average x-coordinates: \frac{0 + 8 + 13}{3} = \frac{21}{3} = 7
2. Average y-coordinates: \frac{-3 + 0 - 7}{3} = \frac{-10}{3}
3. Coordinates of the centroid: (7, -10/3)
4. Verification by drawing medians of each side.