Answer:
Proof below
Step-by-step explanation:
To prove DE and AB are parallel, we compare the slopes of both segments
Suppose we know a line passes through points A(x1,y1) and B(x2,y2). The slope can be calculated with the equation:
[tex]\displaystyle m=\frac{y_2-y_1}{x_2-x_1}[/tex]
Segment AB goes through the points (0,0) and (x2,0), thus:
[tex]\displaystyle m_{AB}=\frac{0-0}{x_2-0}=0[/tex]
The slope is 0 because the line is horizontal.
Now for the segment DE, the endpoints are
[tex]\displaystyle (\frac{x_1}{2},\frac{y_1}{2}),\ (\frac{x_1+x_2}{2},\frac{y_1}{2})[/tex]
The slope is:
[tex]\displaystyle m_{DE}=\frac{\frac{y_1}{2}-\frac{y_1}{2}}{\frac{x_1+x_2}{2}-\frac{x_1}{2}}[/tex]
[tex]\displaystyle m_{DE}=\frac{0}{\frac{x_2}{2}}=0[/tex]
Segment DE is also horizontal, thus is parallel to segment AB.
