How To Calculate the Lengths of the Medians of a Triangle, Given the Three Sides' Lengths

• The medians in a triangle

• Definition: A median of a triangle is a line segment connecting a vertex of the triangle to the midpoint of the opposite side.
• Every triangle has three medians that meet at a single center point, called centroid.
• 

  The three medians of the triangle

• In the triangle △ABC m_a is the median connecting the vertex A to the middle of side BC, m_b is the median connecting the vertex B to the middle of side AC and m_c is the median connecting the vertex C to the middle of the side AB. All the medians intersect at point G, the centroid.
• The point G divides the lengths of the three medians as follows:

• ^AG/_AM = ^BG/_BP = ^CG/_CN = ²/₃
• ^GM/_AM = ^GP/_BP = ^GN/_CN = ¹/₃
• ^GM/_AG = ^GP/_BG = ^GN/_CG = ¹/₂
  

• Apollonius' Theorem

• In a triangle △ABC, for which we know the lengths of the sides "a", "b" and "c", having the opposite vertices A, B and C, respectively, there is a relationship between the lengths of the sides and the lengths of the medians m_a, m_b and m_c, where m_a is drawn from vertex A to side "a", m_b is drawn from vertex B to side "b", and m_c is drawn from vertex C to side "c":

• b² + c² = 2 × (m_a² + (^a/₂)²) =>
• m_a² = ^{(b2 + c2)}/₂ - (^a/₂)²
• a² + c² = 2 × (m_b² + (^b/₂)²) =>
• m_b² = ^{(a2 + c2)}/₂ - (^b/₂)²
• a² + b² = 2 × (m_c² + (^c/₂)²) =>
• m_c² = ^{(a2 + b2)}/₂ - (^c/₂)²

» Apollonius' Theorem and the Medians in a Triangle. Formula. Proof

• Example no. 1

• Two of the three medians should be equal in an Isosceles triangle. Let's calculate these three medians m_a, m_b and m_c and see if two of them are equal or not. The sides' lengths are given: a = 10, b = 6 and c = 10.
• Apply the Apollonius' Theorem and solve for the medians:

We can see that two of the medians are equal, the ones corresponding the congruent sides, m_a ≅ m_c

• Example no. 2

• The length of a median in a right triangle with an angle of 30 degrees, corresponding to the hypotenuse, should be half the length of the hypotenuse. The side lengths in this triangle are given as follows: a = 10 (hypotenuse), b = 5, and c = 8.66025. Let's calculate the median m_a of this right triangle, corresponding to the hypotenuse, and check the above mentioned rule.
• Apply the Apollonius' Theorem and solve for the medians:

We can see tthe length of the median corresponding to the hypotenuse, m_a, is half the legth of the hypotenuse.

• Example no. 3

• Let's calculate the medians m_a, m_b and m_c of a triangle if all the sides' lengths are given: a = 10, b = 6 and c = 10.
• Apply the Apollonius' Theorem and solve for the medians: