Answer:
Segments DE and BC have equal slopes, showing that segments DE and BC are parallel
Step-by-step explanation:
Here we have the coordinates as follows
The coordinates of A is (4, 6)
The coordinates of B is (2, -2)
The coordinates of C is (-2, -4)
Therefore, the coordinates of D the midpoint AB is ((4 + 2)/2, (6 - 2)/2) which gives;
The coordinates of D is (3, 2)
Similarly, the coordinates of E the midpoint AC is ((4 - 2)/2, (6 - 4)/2) which gives;
The coordinates of E is (1, 1)
To prove that segment DE is parallel to segment BC, e show that the slopes of the two segments are equal as follows;
![Slope \, of \, a \, segment = \frac{Change \, in \, the\ y \, coordinates}{Change \, in \, the\, x \, coordinates}](https://tex.z-dn.net/?f=Slope%20%5C%2C%20of%20%5C%2C%20a%20%5C%2C%20segment%20%3D%20%5Cfrac%7BChange%20%5C%2C%20in%20%5C%2C%20the%5C%20y%20%5C%2C%20coordinates%7D%7BChange%20%5C%2C%20in%20%5C%2C%20the%5C%2C%20x%20%5C%2C%20coordinates%7D)
![Slope \, of \, segment \ DE =\frac{2 - 1}{3-1} = \frac{1}{2}](https://tex.z-dn.net/?f=Slope%20%5C%2C%20of%20%5C%2C%20segment%20%5C%20%20DE%20%3D%5Cfrac%7B2%20-%201%7D%7B3-1%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D)
![Slope \, of \, segment \ BC =\frac{-2 - (-4)}{2-(-2)} = \frac{2}{4} =\frac{1}{2}](https://tex.z-dn.net/?f=Slope%20%5C%2C%20of%20%5C%2C%20segment%20%5C%20%20BC%20%3D%5Cfrac%7B-2%20-%20%28-4%29%7D%7B2-%28-2%29%7D%20%3D%20%5Cfrac%7B2%7D%7B4%7D%20%3D%5Cfrac%7B1%7D%7B2%7D)
Therefore, the slopes of segments DE and BC are equal, which shows that segment DE is parallel to BC.