Find the lengths

Given: The vertices of the triangle are A (0, 0, 6), B (0, 4, 0) and C (6, 0, 0).

⇒ x₁ = 0, y₁ = 0, z₁ = 6; x₂ = 0, y₂ = 4, z₂ = 0; x₃ = 6, y₃ = 0, z₃ = 0

We know that the median is a line segment through a vertex of a triangle to the midpoint of the side opposite to the vertex.

So, let the medians of this triangle be AD, BE and CF corresponding to the vertices A, B and C respectively.

⇒ D, E and F are the midpoints of the sides BC, AC and AB respectively.

By Midpoint Formula, we know that the coordinates of the mid-point of the line segment joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) are .

So, we have

The coordinates of D = (3, 2, 0)

The coordinates of E = (3, 0, 3)

And the coordinates of F = (0, 2, 3)

By Distance Formula, we know that the distance between two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) is given by .

The lengths of the medians are:

So, the lengths of the medians of the given triangle are 7, and 7.