Triangle Medians and Centroids

Triangle Medians

• A median of a triangle:
  • A segment that connects a vertex of the triangle to the midpoint of the opposite side.

Medians in Practice

• Example given: Triangle AYWT with medians WG, TD, and YF.
  • Midpoints:
  • Point P is the midpoint of WH.
  • Point R is the midpoint of TH.
  • Point K is the midpoint of YH.
• Relationships between segments:
  • If segment DH is compared to segment TH, then the relationship is:
  • $TH = 2 imes DH$
  • For segments YH and FH:
  • $YH = 2 imes FH$
  • For segments WH and GH:
  • $WH = 2 imes GH$

Centroid of a Triangle

• Centroid:
  • The meeting point of the three medians of a triangle.
  • A conjecture can be made about the distances related to the centroid and the triangle’s vertices:
  • The centroid is located at a distance that is double from the vertex compared to its distance to the midpoint of the opposite side.

Midsegment Theorem of Triangle

• The midsegment of a triangle is defined as follows:
  • It connects the midpoints of two sides of a triangle.
  • Its length is:
  • ext{Length of Midsegment} = rac{1}{2} imes ext{Length of Third Side}
  • It is parallel to the third side of the triangle.

Application of Midsegment Theorem

• Problem outlined: Proving that segments SG and HT are parallel when SH = HC and GT = TC:
  • By definition:
  • SH and HC are midpoints, thus HT is a midsegment.
  • Therefore, $SG || HT$

Centroid Calculations for Coordinates

• Given Triangle AMBG with vertices:
  • M(0, -3), B(8, 0), and G(13, -7).
• Centroid Coordinates Calculation:
  • Average of the x-coordinates:
  • rac{0 + 8 + 13}{3} = rac{21}{3} = 7
  • Average of the y-coordinates:
  • rac{-3 + 0 - 7}{3} = rac{-10}{3} ext{ which is approximately } -3.33
  • Thus, the coordinates of the centroid are:
  • $ext{Centroid} ext{ at } (7, -3.33)$
  • Verifying centroid by drawing the medians from each vertex to the midpoints of the opposite sides.