Hello, Jenny!

1) Use vectors to prove that the line segment joining the midpoint of two sides of a triangle

is parallel to the third side and half as long. Code:

A
*
* *
* *
D *--------* E
* *
* *
*-----------------*
B C

Let $\displaystyle D$ and $\displaystyle E$ be the midpoints of $\displaystyle AB$ and $\displaystyle AC$, respectively.

Draw line segment $\displaystyle DE.$

We know that: .$\displaystyle \overrightarrow{BA} + \overrightarrow{AC} \:=\:\overrightarrow{BC}$ **[1]**

We know that: .$\displaystyle \overrightarrow{DA} + \overrightarrow{AE} \:=\:\overrightarrow{DE}$ **[2]**

We are told that: .$\displaystyle \overrightarrow{DA} \:=\:\frac{1}{2}\overrightarrow{BA}$ .and .$\displaystyle \overrightarrow{AE} \:=\:\frac{1}{2}\overrightarrow{AC}$

Substitute into **[2]**: .$\displaystyle \overrightarrow{DE} \:=\:\frac{1}{2}\overrightarrow{BA} + \frac{1}{2}\overrightarrow{AC} \:=\:\frac{1}{2}\left(\overrightarrow{BA} + \overrightarrow{AC}\right)$

From **[1]**, we have: .$\displaystyle \overrightarrow{DE} \:=\:\frac{1}{2}\overrightarrow{BC}$

Therefore: .$\displaystyle \overrightarrow{DE} \,\parallel\, \overrightarrow{BC}$ .and .$\displaystyle \left|\overrightarrow{DE}\right| \:=\:\frac{1}{2}\left|\overrightarrow{BC}\right|$