Question

A,B and C are the vertices or a triangle.

A has the 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.

Answer 1

The slopes of DE and BC are the same: 0.5. The two lines are therefore parallel.

Explanation:

See the attached graph, made in DESMOS.

The midpoints D and E may be determined by taking the average values of x and y that form the endpoints of lines AB (midpoint D) and AC (midpoint E):

Midpoint Endpoint1 Endpoint 2 Midpoint

D A(4,6) B(2,-2) D(3,2) [(4+2)/2 =3 and (6-2)/2 = 2]

E A(4,6) C(-2,-4) E(1,1) [(4-2)/2 =1 and (6-4)/2 = 1]

If DE is parallel to BC, both lines must have the same slope. Find the slopes for DE and BC by calculating the Rise/Run (the change in y over the cahnge in x):

Slope of DE: Rise = (2-1) = 1, Run = (3-1) = 2. Slope = (1/2) or 0.5

Slope of BC: Rise = (-2-(-4)) = 2, Run = (2-(-2)) = 4. Slope = (1/2) or 0.5

The slopes of DE and BC are the same: 0.5. The two lines are therefore parallel.