(-5, 2), (3, -6) and (7, 4) are the vertices of a triangle. Find the length of its median through the vertex (3, -6).

Let A = (3, -6), B = (-5, 2) and C = (7, 4).

From figure, AD is the median.

Since, AD is median so, BD = DC.

Thus, D is mid-point of BC.

By formula,

Mid-point = $\Big(\dfrac{x$1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

Substituting value we get,

$D = \Big(\dfrac{-5 + 7}{2}, \dfrac{2 + 4}{2}\Big) \\[1em] = \Big(\dfrac{2}{2}, \dfrac{6}{2}\Big) \\[1em] = (1, 3).$

Distance between two points = $\sqrt{(x$2 - x1)^2 + (y2 - y1)^2}

Substituting values we get,

$AD = \sqrt{(1 - 3)^2 + [3 - (-6)]^2} \\[1em] = \sqrt{(-2)^2 + (9)^2} \\[1em] = \sqrt{4 + 81} \\[1em] = \sqrt{85} \\[1em] = 9.22$

Hence, the length of its median through the vertex (3, -6) = 9.22 units.