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Triangles 🔺️: Centroid and Medians

Definition of a Median of a Triangle

  • A median is a line segment drawn from a vertex of a triangle to the midpoint of the opposite side.

Properties of Medians

  • Each median divides the triangle into two smaller triangles of equal area.
  • Medians meet at a point called the centroid, which is the center of mass of the triangle.

Centroid of a Triangle

  • The centroid (denoted as G) is the intersection point of the three medians.
  • Important property: The distance from each vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side.

Example:

  1. If Vertex A’s distance to centroid G is d_A,
  2. Distance from centroid G to midpoint M of side BC is rac{d_A}{2}.

The Relationship Between Segments and Midpoints

  • When determining relationships among segments in a triangle formed by midpoints:
    • Given segments are often related multiplicatively based on their connections to midpoints.
      • Example: In triangle AYWT:
      • If DH is segment-1, and TH is segment-2: TH = 2 imes DH.
      • If FH is segment-3, and YH is segment-4: YH = 2 imes FH.
      • Thus, if segment GH is considered: WH = 2 imes GH.

Triangle Midsegment Theorem

  • Definition: The midsegment of a triangle connects the midpoints of two sides of the triangle.
  • Properties:
    • Length of the midsegment is half the length of the third side.
    • The midsegment is parallel to the third side.

Proving Properties of Midsegments

  1. Given that SH = HC and GT = TC, point H and point T are midpoints.
  2. Therefore, HT is the midsegment of triangle ASGC, leading to the conclusion that SG || HT.

Finding the Centroid of a Triangle

Given Points:

  • Vertices of triangle AMBG are M(0, -3), B(8, 0), and G(13, -7).

Steps to Find Centroid:

  1. Average of the x-coordinates:
    x = \frac{0 + 8 + 13}{3} = \frac{21}{3} = 7
  2. Average of the y-coordinates:
    y = \frac{-3 + 0 - 7}{3} = \frac{-10}{3} = -\frac{10}{3}
  3. Coordinates of the Centroid:
    • Therefore, the centroid G is located at (7, -10/3).
  4. Verification:
    • Verify by drawing medians from each vertex to the opposite side to visually confirm the calculated centroid.

Conclusion

  • The centroid serves as a crucial point in progressive geometric calculations and assists in understanding the relationships present in triangle geometry.
  • The relationships among midpoints and segment lengths provide foundational insights into the structure of triangles and their properties.