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.