The term "median of a triangle" is still in the course of geometry of the 7th grade, however, its finding causes some difficulties for pupils who finish school and for their parents.
In this article, the method by which you can find the median of an arbitrary triangle will be described compactly.
You will need
To begin with, you should determine the concept of the median (find out what it means).
Look at the arbitrary triangle ABC. The BD-segment, which connects the vertex of the triangle to the middle of the opposite side, is the median.
Thus, thanks to the above definition and accompanying figure 1, you should understand that any triangle has 3 medians that intersect within this figure.
The point of intersection of the medians is the center of gravity of the triangle, or, as it is also called, the center of mass. Each median is divided by the point of intersection of medians in a ratio of 2: 1, counting from the top.
Note also the fact that the triangles to which the original triangle will be divided by all their medians have the same area.
In order to calculate the median, you need to use a specially designed algorithm. Formula for calculating the median through the sides of the triangle</a> Looks as shown in Figure 2,
Where m (a) is the median of the triangle ABC connecting the vertex A with the middle of the side BC,
B is the side of the speaker of the triangle ABC,
C - side AB of the triangle ABC,
A is the side of the BC triangle ABC.
From the presented formula it follows that knowing the lengths of all the medians of the triangle, you can find the length of any of its sides.
If you need a formula to find the side of a triangle through its medians, it looks as shown in Figure 3, where:
A is the side of the BC of the triangle ABC,
M (b) is the median emerging from vertex B,
M (c) is the median emerging from vertex C,
M (a) is the median emerging from vertex A.
For the correct calculation of the median, you need to familiarize yourself with special cases that can occur when solving equations with the presence of an arbitrary triangle in them.
1. In an equilateral triangle, the median emerging from the vertex, which is formed by equal sides, is:
- bisector of the angle formed by equal sides of the triangle-
-the height of this triangle-
2. In an equilateral triangle, all medians are equal. All medians are the bisectors of the corners and the heights of this triangle.