Final answer:
The midpoint coordinates of D and E are determined and then used to calculate the slopes of DE and BC. Both segments have the same slope (1/2), which proves that DE is parallel to BC.
Step-by-step explanation:
To prove that line segment DE is parallel to line segment BC, we need to show that they have the same slope, as parallel lines in the Cartesian plane have equal slopes.
Calculation of Midpoints D and E
First, we calculate the midpoint D of AB:
D's x-coordinate = (xA + xB)/2 = (4 + 2)/2 = 3
D's y-coordinate = (yA + yB)/2 = (6 - 2)/2 = 2
So, D has coordinates (3, 2).
Now we calculate the midpoint E of AC:
E's x-coordinate = (xA + xC)/2 = (4 - 2)/2 = 1
E's y-coordinate = (yA + yC)/2 = (6 - 4)/2 = 1
So, E has coordinates (1, 1).
Calculation of Slopes
Slope of DE = (yE - yD)/(xE - xD)
= (1 - 2)/(1 - 3)
= -1/(-2)
= 1/2
Slope of BC = (yC - yB)/(xC - xB)
= (-4 + 2)/(-2 - 2)
= -2/(-4)
= 1/2
Since both slopes are equal, DE is parallel to BC.
the complete Question is given below:
A, B and C are the vertices of a triangle.
A has coordinates (4, 6)
B has coordinates (2, —2)
C has coordinates (—2, —4)
D is the midpoint of AB.
E is the midpoint of AC.
Prove that DE is parallel to BC.
You must show each stage of your working.