A median of a triangle is a segment drawn from one vertex to the midpoint of the opposite side. Every triangle has three medians, and they always meet at one point called the centroid. This point is important because it gives a simple way to locate the triangle's balancing point.
Medians and centroids connect geometric construction, coordinate geometry, and physical ideas like center of mass.
The centroid divides each median in a fixed 2 to 1 ratio, with the longer part between the vertex and the centroid. If a triangle has uniform thickness and density, the centroid is the point where it would balance on a pin. In coordinate geometry, the centroid is found by averaging the x-coordinates and y-coordinates of the three vertices.
This makes the centroid useful for solving geometry problems, modeling shapes, and understanding symmetry and balance.
Key Facts
- A median connects a vertex of a triangle to the midpoint of the opposite side.
- The three medians of any triangle are concurrent, meaning they intersect at one point.
- The point where the medians intersect is the centroid, usually labeled G.
- The centroid divides each median in the ratio 2:1 from the vertex to the midpoint.
- If AM is a median and G is the centroid, then AG = (2/3)AM and GM = (1/3)AM.
- For vertices A(x1, y1), B(x2, y2), and C(x3, y3), the centroid is G = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3).
Vocabulary
- Median
- A median is a segment from a vertex of a triangle to the midpoint of the opposite side.
- Centroid
- The centroid is the common intersection point of the three medians of a triangle.
- Midpoint
- A midpoint is the point that divides a segment into two equal lengths.
- Concurrent
- Concurrent lines or segments are three or more lines or segments that all pass through the same point.
- Center of Mass
- The center of mass is the point where an object's mass can be treated as concentrated for balance and motion.
Common Mistakes to Avoid
- Confusing a median with an altitude. A median goes to the midpoint of the opposite side, while an altitude meets the opposite side at a right angle.
- Placing the centroid halfway along a median. The centroid is not the midpoint of a median because it divides the median in a 2:1 ratio from the vertex.
- Using only one coordinate instead of averaging all three vertices. The centroid formula requires the mean of all three x-values and the mean of all three y-values.
- Assuming the centroid can lie outside the triangle. The centroid of a triangle is always inside the triangle because all three medians meet in the interior.
Practice Questions
- 1 In triangle ABC, D is the midpoint of BC and AD is a median. If AD = 18 cm and G is the centroid, find AG and GD.
- 2 Find the centroid of the triangle with vertices A(2, 5), B(8, 1), and C(-1, 6).
- 3 A triangular cardboard cutout has uniform thickness and density. Explain why the centroid is the point where the triangle balances, and describe how the three medians help locate it.