Question

Prove that DE is parallel to BC?

Answer 1

Please see below.

Explanation:

We use two concepts here.

First that midpoint of segment joining two points `(x_1,y_1)` and `(x_2,y_2)` is `((x_1+x_2)/2,(y_1+y_2)/2)` and

Using this coordinates of `D` are `((4+2)/2,(6-2)/2)` i.e. `(3,2)`

and coordinates of `E` are `((4-2)/2,(6-4)/2)` i.e. `(1,1)`

Second that slope of line joining two points `(x_1,y_1)` and `(x_2,y_2)` is `(y_2-y_1)/(x_2-x_1)`

As coordinates of `B` and `C` are `(2,-2)` and `(-2,-4)`, the slope `BC` is `(-4-(-2))/(-2-2)=(-2)/(-4)=1/2`

and as coordinates of `D` and `E` are `(3,2)` and `(1,1)`, the slope `BC` is `(1-2)/(1-3)=(-1)/(-2)=1/2`

As slope of `DE` and `BC` are equal, `DE`||`BC`