Answer:
see explanation
Step-by-step explanation:
Parallel lines have equal slopes.
To find D and E use the midpoint formula
Given endpoints (x₁, y₁ ) and (x₂, y₂ ) then the midpoint is
( [tex]\frac{x_{1}+x_{2} }{2}[/tex], [tex]\frac{y_{1}+y_{2} }{2}[/tex] )
Here (x₁, y₁ ) = A(4, 6) and (x₂, y₂ ) = B(2, - 2) , then
D = ([tex]\frac{4+2}{2}[/tex], [tex]\frac{6-2}{2}[/tex] ) = (3, 2 ) and
let (x₁, y₁ ) = B(2, - 2[tex]\frac{-4+2}{-2-2}[/tex] ) and (x₂, y₂ ) = C(- 2, - 4 ), then
E = ( [tex]\frac{4-2}{2}[/tex], [tex]\frac{6-4}{2}[/tex] ) = (1, 1 )
Use the slope formula to find slopes of DE and BC
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = D(3, 2) and (x₂, y₂ ) = E(1, 1), then
[tex]m_{DE}[/tex] = [tex]\frac{1-2}{1-3}[/tex] = [tex]\frac{-1}{-2}[/tex] = [tex]\frac{1}{2}[/tex]
Repeat with (x₁, y₁ ) = B(2, - 2) and (x₂, y₂ ) = C(- 2, - 4), then
[tex]m_{BC}[/tex] = [tex]\frac{-4+2}{-2-2}[/tex] = [tex]\frac{-2}{-4}[/tex] = [tex]\frac{1}{2}[/tex]
Since the slopes are equal then DE and BC are parallel lines
