In a triangle, a median is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. Each triangle has three medians, one from each vertex, and they intersect at a single point called the centroid.

Here are some key points about medians:

1. Definition: Each median divides the triangle into two smaller triangles of equal area.

2. Intersection Point (Centroid): The medians intersect at the centroid, which is located 2/3 of the distance from each vertex along the median. Thus, if the median length is $m$, the centroid divides it into two segments: $\frac{2}{3}m$ from the vertex to the centroid and $\frac{1}{3}m$ from the centroid to the midpoint of the side.

3. Finding the Length of the Median: For a triangle with vertices $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$, the median from $A$ to side $BC$ can be calculated using the formula: $m_a = \sqrt{\frac{2(b^2 + c^2) - a^2}{4}}$ where $a, b,$ and $c$ are the lengths of the sides opposite vertices $A, B,$ and $C$, respectively.

4. Properties:

   • The medians of a triangle divide it into six smaller triangles of equal area.
   • The centroid is also the triangle's center of mass, assuming it is made of a uniform material.

5. Applications: The median has applications in geometry, trigonometry, and physics (especially in problems involving centers of mass).

Would you like to see an example calculation of a median, or do you have any specific questions?

Related Questions:

1. How do you derive the formula for the length of a median in a triangle?
2. What is the relationship between the centroid and other centers in a triangle (circumcenter, orthocenter, incenter)?
3. How does the centroid divide each median, and why is this ratio always 2:1?
4. Can medians of a triangle be used to calculate its area?
5. What are the properties of medians in an isosceles triangle?

Tip:

The centroid of a triangle not only divides each median into a 2:1 ratio but also balances the triangle in terms of mass distribution, making it the triangle’s center of gravity.