The length of a median of a triangle
This lesson is focused on the formula expressing the length of a median of a triangle.
The proof of the formula is presented, which is based on the
Law of cosines (see the lesson
Proof of the Law of Cosines revisited under the topic
Trigonometry of
the section
Algebra-II in this site). The examples are provided too, showing how to use this formula.
Theorem 1
In a triangle with the sides
a,
b and
c the median drawn to the side
c has the length of
=
. Prove.
Proof
Figure 1 shows the triangle ABC with the sides a (BC), b (AC) and c (AB) and
the median m (CD) drawn to the side c. Let d be the length of the segment AD
and e be the length of the segment DB the median CD divides the side AB. Denote
as  and  the angles at the intersection the median CD and the side AB.
Apply the Law of Cosines to express the length of the side AC of the triangle ADC
 =  .
Apply the Law of Cosines to express the length of the side BC of the triangle BDC
 =  .
|
Figure 1. To the Theorem 1
|
Now, take the sum of the last two equalities. Note that
, because the angles
and
are supplementary angles. Also note that
, because
CD is
the median. Therefore, when taking the sum, the terms
and
cancel each other out. So, after summing we get
+
=
+
+
.
Substitute here
and
using the fact that
CD is the median. We get
+
=
+
+
,
or
= (
+
-
)/4.
This is what to be demonstrated. The proof is completed.
Theorem 2
In a triangle with the sides
a,
b, and
c the lengths of the medians
,
and
, drawn to the sides
a,
b and
c respectively satisfy to the identity
=
. Prove.
Proof
Based on the
Theorem 1 we have three equalities
=
,
=
, and
=
.
By taking the sum of these equalities, we get the identity
=
.
This is exactly what to be demonstrated. The
Theorem 2 is proved.
Example 1
In the triangle the side lengths are a = 5, b = 6 and c = 4 (Figure 2).
Find the length of the median drawn to the side c.
Solution
Apply the formula for the median length
= =  =  =
 =  .
Answer. = =~ 5.148 (approximately).
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Figure 2. To the Example 1
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Example 2
In the isosceles triangle the lateral side has the length of 4.
The median drawn to the lateral side has the length of 3 (Figure 3).
Find the length of the base of the triangle.
Solution
Let  be the length of the base of the triangle.
Apply the formula for the median length. You get the equation
 .
Simplify this equation step by step as shown below:
 ,
 ,
 ,
 ,
 ,
 .
Answer. The base of the triangle has the length of  .
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Figure 3. To the Example 2
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