Height in a triangle

In this article, we will learn everything you need to know about medians in a triangle! Don't worry, the material about medians in a triangle is easy to understand and very logical.

A median in a triangle is a line segment that extends from a vertex to the midpoint of the opposite side, dividing it into two equal parts.
You can remember that "median" in real life represents the middle point, and similarly here it divides the side in the middle!

We can see in the drawing:

In triangle $ABC$
$AD$ is a median - extends from vertex $A$ and divides the opposite side $CB$ into two
equal parts: $CD=BD$

Additional Properties of a Median in a Triangle:

1. In every triangle, it is possible to draw 3 medians.
2. All 3 medians intersect at one point.
3. The median to a side of a triangle creates 2 triangles of equal area.

You can see for yourself:

Since there are 3 vertices in a triangle, there can be 3 medians.
Each median extends from a vertex to the opposite side and bisects it.
All medians intersect at one point.

Reminder:
How do we calculate the area of a triangle?

$height*corresponding~side~to~height \over 2$

If we take for example the triangle $ABC$ and want to calculate its area when:
$AD$ height = $6$
$BC = 8$

We get that the area of triangle $ABC$ is:
$\frac{6*8}{2}=24$
Now if we draw median $AD$ we will see that the two triangles it creates are equal in area.
The side is divided in the middle so it is identical in both triangles and the height is identical.
Therefore, the area of each created triangle is identical and will be equal to half the area of triangle $ABC$

$\frac{6*4}{2}=12$

1. In an equilateral triangle - the median is also a height and an angle bisector.
   As shown in the figure:
   
   $AD$ is a height to side $CB$
   and also a median to side $CB$ (divides it into two equal parts)
   and also an angle bisector

$A$

1. In an isosceles triangle - the median drawn from the vertex angle is also a height and an angle bisector.
   Let's see in the figure:
   

$AD$ is a median drawn from vertex angle $A$.
It is also a height to side $CB$, also a median to $CB$, and also bisects the vertex angle $A$.

3. In a right triangle - the median to the hypotenuse equals half the hypotenuse.

We can see in the figure:

Triangle $ABC$ is a right triangle.
$CD$ is the median to the hypotenuse and equals half of the hypotenuse.
That is
$CD=AD=DB$

Given:

$CD$ is a median in triangle $ABC$
$CE$ is a median in triangle $ACD$

$AB=14$

1. Calculate the length of segment $AE$.
2. What is the area of triangle $ECD$ if it is known that the area of triangle $ACE$ is $6$?
3. Based on your answer in part b, what is the area of triangle $DCB$?

Solution:

1. Let's mark the given information on the drawing.
   

Given that – $AB =14$
Since $CD$ is a median,
$AD=DB=7$
because the median bisects the side at its midpoint.
Given that $CE$ is also a median.
Therefore $AE=ED=3.5$.

2. Given that the area of triangle $ACE$ is $6$
therefore the area of triangle $ECD$
must also be $6$. A median divides the triangle into two triangles of equal area.

3. The area of triangle $DCB$ must be equal to the area of triangle $ACD$.

Triangle $ACD$ consists of two triangles with equal areas that sum up to $12$.
Therefore, the area of triangle $DCB$ is $12$.