The co-ordinates of vertices of triangle are (2, –4), (6, –2) and (–4, 2) respectively. Let us find the length of three medians of triangle.

The co-ordinates of vertices of a triangle ABC are A(2, – 4), B(6, –2) and C (– 4, 2) respectively.

To calculate, the length of Median AD, first we’ll calculated the coordinates of mid-point (d) of BC.

Let the coordinates of that mid-point be (x, y) –

And, we know mid-point formula, i.e. the coordinates of mid-point of line joining (x₁, y₁) and (x₂, y₂) is

⇒ x = 1 and y = 0

⇒ the coordinates of one end of median(x₁, y₁) = (2, – 4) and of another end(x₂, y₂) = (1, 0).

Now, we know the length = √ ((x₂ – x₁)² + (y₂ – y₁)²)

⇒ Length of median = √ ((1 –(2))² + (0 –(– 4))²)

⇒ Length of median = √ (1 + 16)

⇒ Length of median = √ 17

Now, to calculate, the length of Median BE, first we’ll calculated the coordinates of mid-point (E) of AC.

Let the coordinates of that mid-point be (x, y) –

⇒ x = – 1 and y = – 1

⇒ The coordinates of one end of median(x₁, y₁) = (6, – 2) and of another end(x₂, y₂) = (– 1, – 1).

Now, we know the length = √ ((x₂ – x₁)² + (y₂ – y₁)²)

⇒ Length of median = √ ((– 1 –(6))² + (– 1 –(– 2))²)

⇒ Length of median = √ (49 + 1)

⇒ Length of median = √ 50

And, now To calculate, the length of Median CG, first we’ll calculated the coordinates of mid-point (G) of AB.

Let the coordinates of that mid-point be (x, y) –

⇒ x = 4 and y = – 3

⇒ The coordinates of one end of median(x₁, y₁) = (– 4, 2) and of another end(x₂, y₂) = (4, – 3).

Now, we know the length)

⇒ Length of median

⇒ Length of median

⇒ Length of median