medians

Median of Triangle Formulas

The formula for the first median of a triangle which form on the side a is represented as follows: $M_a=\frac{\sqrt{2b^2+2c^2-a^2}}{4}$

 Where,

 $M_a$  is the median of the triangle ABC.

 a, b and c are called a triangle’s sides. 

The formula for the second median of a triangle which form on the side a is represented as follows: $M_b=\frac{\sqrt{2a^2+2c^2-b^2}}{4}$

 Where,

 $M_b$ is the median of the triangle ABC.

 a, b and c are called a triangle’s sides. 

The formula for the third median of a triangle which form on the side a is represented as follows: $M_c=\frac{\sqrt{2a^2+2b^2-c^2}}{4}$

 Where,

$M_c$ is the median of the triangle ABC.

  a, b and c are called a triangle’s sides.

How to find the Median of Triangle with Coordinates?

If the coordinates of the three vertices of a triangle are provided, then the following steps are used to find the length of the median of the triangle.

Step 1: We use the provided coordinates of the vertices of the triangle and find the coordinates of the midpoint of the line segment on which the median is formed. The coordinates of the midpoint of the line segment can be obtained by using the midpoint following formula.

The Coordinate midpoint of the line segment = $\left(\frac{p_1+p_2}{2},\frac{q_1+q_2}{2}\right)$

Where $(p_1,q_1)$ and $(p_2,q_2)$ are the coordinates of the endpoints of the line segment.

Step 2: After getting the coordinates of the midpoint, we can find the length of the median using the distance formula. One endpoint is the vertex from where the median starts and the other is the midpoint of the line segment where the median lies. 

Step 3: The length of the median can be found with the help of the following distance formula.

The distance between two point = $\sqrt{(r_2-r_1{)}^2+(s_2-s_1{)}^2}$

where, $(r1,s1)$ and $(r2,s2)$ are the coordinates of both ends of the median.

Example: Find the length of the median AD if the coordinates of the triangle ABC are given as, $A(6,10),B(8,2),C(-8,6)$

Solution: Let the point D which is the midpoint of side BC on which the median is formed.

 The coordinates of point $D=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ --- formula no.01.

 As given, the coordinates of $B=(8,2)$ and $C=(-8,6)$. 

 Substituting the values of B and C in the formula no.01, we get.

$\left[\frac{8+(-8)}{2},\frac{2+6}{2}\right]$ Therefore, the coordinates of points D as $(0,4$). 

 Now, the length of the median, AD = \sqrt{(x_3 - x_4)^2 + (y_3 - y_4)^2} \tag{Formula 2}
 where the coordinates of the median are A (6, 10), and D (0, 4). Substituting the values in the formula no.02, we get.

$AD=\sqrt{(0-6{)}^2+(4-10{)}^2}$

 = $AD=\sqrt{36+36}$

 =$\sqrt{72}$ 

 = 8.48units.

 Thus, the length of the median AD is 8.46 units.