Medians, Angle Bisectors, and Altitudes of a Triangle
Grade 7 Geometry — Three Special Segments and Their Remarkable Properties
Introduction
Every triangle, no matter its shape or size, contains three special types of segments:
3 Medians
Vertex → Midpoint of opposite side
3 Angle Bisectors
Bisects the angle at each vertex
3 Altitudes
Vertex → Perpendicular to opposite side
Each type connects a vertex to the opposite side in a specific way. These segments reveal deep properties of triangles: their intersection points are important triangle centers, used in constructions, proofs, and real-world applications. They form the foundation for advanced geometry.
Part 1: Medians of a Triangle
Definition: Median
A median of a triangle is a segment connecting a vertex to the midpoint of the opposite side. Every triangle has exactly three medians.
The median from vertex $A$ to the midpoint $M$ of side $\overline{BC}$ is denoted $\overline{AM}$ or $m_a$.
Key Properties of Medians
- Divides the opposite side into two equal parts (that's the midpoint!).
- Does NOT necessarily bisect the angle at the vertex.
- All three medians lie entirely inside the triangle.
- A median divides the triangle into two smaller triangles of equal area (same base length, same height).
The Centroid (G)
Centroid: The point where all three medians intersect. Denoted G.
✅ All three medians always meet at a single point — a remarkable theorem!
✅ The centroid always lies inside the triangle (acute, right, or obtuse).
✅ The centroid is the center of mass (balance point) of the triangle.
⭐ The 2:1 Rule
The centroid divides each median in the ratio $\mathbf{2:1}$ from the vertex:
$$\frac{AG}{GM} = \frac{2}{1}$$
The centroid is twice as far from the vertex as it is from the midpoint of the opposite side.
📐 Centroid Coordinates
If the vertices are $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$, then:
$$G = \left(\frac{x_1 + x_2 + x_3}{3},\; \frac{y_1 + y_2 + y_3}{3}\right)$$
The centroid is the average of all three vertex coordinates!
📏 Median Length Formula
The length of the median from vertex $A$ to side $\overline{BC}$ (of length $a$):
$$m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}$$
Interactive: Medians and Centroid
Drag vertices A, B, C to see how medians and centroid change. Notice: G always stays inside!