Calculate the length of the median through the vertex A of the triangle ABC with vertices A(7, -3), B(5, 3) and C(3, -1).

Below figure shows the triangle ABC with vertices A(7, -3), B(5, 3) and C(3, -1):

Let D (x, y) be the mid-point of BC, then AD is the median through A.

As D is the mid-point of BC, then coordinates of D by mid-point formula are

$\Rightarrow x = \dfrac{5 + 3}{2} \text{ and } y = \dfrac{3 + (-1)}{2} \\[1em] \Rightarrow x = \dfrac{8}{2} \text{ and } y = \dfrac{2}{2} \\[1em] \Rightarrow x = 4 \text{ and } y = 1.$

Hence, coordinates of D are (4, 1).

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

By distance formula length of median,

$AD = \sqrt{(7 - 4)^2 + (-3 - 1)^2} \\[1em] = \sqrt{3^2 + (-4)^2} \\[1em] = \sqrt{9 + 16} \\[1em] = \sqrt{25} \\[1em] = 5.$

Hence, the length of the median through vertex A is 5 units.