3. find the midsegment of triangle bcd, which is parallel to bc. label the endpoints of the midsegment as e and g. what are the coordinates of e and g? show work verifying that eg and bc are

Since we want the midsegment that's parallel to BC, the endpoints will be the
midpoints of BD and CD. So 
Midpoint BD: 
((-3 + -1)/2, (1 + -3)/2) = (-4/2, -2/2) = (-2,-1)  
Midpoint CD:
 ((3+ -1)/2, (3 + -3)/2) = (2/2, 0/2) = (1, 0)  
So the coordinates of E & G are (-2,-1) and(1,0) respectively.  
Segment EG will be parallel to segment BC if they have the same slope. So 
Slope BC: 
(3 - 1)/(3 - -3) = 2/6 = 1/3  
Slope EG 
(0 - -1)/(1 - -2) = 1/3  
Since both EG and BC have the same slope, they are parallel.