Median

Definition

In geometry, a median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side. This key feature of triangles divides them into two smaller triangles of equal area and plays an important role in understanding the balance and symmetry within the shape. Medians are essential when discussing properties of triangles, such as their centers, and are closely related to the concepts of bisectors and altitudes.

5 Must Know Facts For Your Next Test

1. Every triangle has three medians, one from each vertex to the opposite side's midpoint.
2. The centroid divides each median into two segments, with the segment connecting the vertex to the centroid being twice as long as the segment from the centroid to the midpoint.
3. Medians can be used to prove various properties of triangles, including the triangle area and relationships among angles.
4. In an equilateral triangle, all medians are also altitudes and angle bisectors, showcasing its symmetrical properties.
5. The length of a median can be calculated using the formula: $$m_a = \sqrt{\frac{2b^2 + 2c^2 - a^2}{4}}$$, where 'm_a' is the length of the median to side 'a', and 'b' and 'c' are the lengths of the other two sides.

Review Questions

• How does the median of a triangle relate to its areas and overall balance?
  • The median connects a vertex to the midpoint of the opposite side, creating two smaller triangles with equal areas. This property highlights how medians contribute to the balance within a triangle. The centroid, where all three medians intersect, further emphasizes this balance as it acts as the center of mass for the triangle.
• Explain how medians differ from altitudes in terms of their geometric properties.
  • While both medians and altitudes are segments drawn from a vertex to the opposite side, they serve different purposes. A median connects a vertex to the midpoint of the opposite side, splitting it into equal-area triangles. In contrast, an altitude is perpendicular to the opposite side, determining height and affecting area calculations. Understanding these distinctions helps clarify their unique roles in triangle geometry.
• Evaluate how understanding medians can help solve complex problems involving triangle congruence and symmetry.
  • Understanding medians is crucial for solving problems related to triangle congruence and symmetry. By recognizing that medians create equal areas within triangles, one can apply this knowledge to establish relationships between different triangles or even deduce properties about unknown angles or sides. Furthermore, since all medians intersect at the centroid, this central point can serve as a reference for proving congruence in overlapping or equilateral triangles, highlighting how key features interconnect in geometric proofs.

Related terms

Centroid:

The point where all three medians of a triangle intersect, often referred to as the center of mass or balance point.

Altitude:

A perpendicular segment from a vertex of a triangle to the line containing the opposite side, which helps determine the height and area of the triangle.

Midpoint: The point that divides a line segment into two equal parts, essential for constructing medians in triangles.