Mathematics

Problem

Consider

\triangle ABC

with

A = (\mathbf{a})

B = (\mathbf{b})

and

C = (\mathbf{c})

. If

\mathbf{b} \cdot (\mathbf{a} + \mathbf{c}) = \mathbf{b} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}

|\mathbf{b} - \mathbf{a}| = 3

and

|\mathbf{c} - \mathbf{b}| = 4

, then the angle between the medians

AM

and

BD

is a.

\pi - \cos^{-1}\left(\frac{1}{5\sqrt{13}}\right)

\pi - \cos^{-1}\left(\frac{1}{13\sqrt{5}}\right)

\cos^{-1}\left(\frac{1}{5\sqrt{13}}\right)

\cos^{-1}\left(\frac{1}{13\sqrt{5}}\right)