Question

Use vectors to prove the following theorems from geometry:

The line segment joining the midpoints of two sides of any triangle is parallel to the third side and half its length.

Short answer

We use the rules of adding vectors to prove this statement.

Step-by-step solution

Concept and formula used

Consider $\mathit{R}\mathbf{=}\mathit{A}\mathbf{+}\mathit{B}$, the points${\mathit{m}}_{\mathbf{a}}$and${\mathit{m}}_{\mathbf{b}}$are the midpoints of the vectors$\mathit{A}$and$\mathit{B}$respectively and the point '$\mathit{O}$' is taken to be the origin.

Usage of Vector in Geometry

The vector from to is

$M_a=\frac{A}{2}$

$$\begin{gathered} M_b=R-\frac{B}{2} \\ =A+\frac{B}{2} \end{gathered}$$

$$\begin{gathered} J=M_b-M_a \\ =A+\frac{B}{2}-\frac{A}{2} \\ =\frac{A+B}{2} \\ J=\frac{R}{2} \end{gathered}$$

We can deduce that the line drawn between the two triangle sides' midpoints is parallel to the third side and half its length.

We use the rules of adding vectors to prove this statement.