Answer:
The equation of the line of reflection is [tex]y = 2\cdot x +1[/tex].
Step-by-step explanation:
After looking carefullt the figure, we notice that line of reflection passes through B and midpoints in line segments AA' and CC'. If we know that [tex]A(x, y) = (-1, 4)[/tex], [tex]A'(x, y) = (3, 2)[/tex], [tex]C(x, y) = (-4, -2)[/tex] and [tex]C'(x,y) = (0, -4)[/tex], the midpoints of each line segment is:
Line segment AA'
[tex]A'' = \left(\frac{-1+3}{2},\frac{4+2}{2}\right)[/tex]
[tex]A'' = (1,3)[/tex]
Line segment CC'
[tex]C'' = \left(\frac{-4+0}{2}, \frac{-2-4}{2} \right)[/tex]
[tex]C'' = (-2, -3)[/tex]
From Analytical Geometry we know that any linear function can be found by knowing two distinct points. The standard form of the linear function is represented by:
[tex]y = m\cdot x+b[/tex]
Where:
[tex]x[/tex] - Independent variable, dimensionless.
[tex]y[/tex] - Dependent variable, dimensionless.
[tex]m[/tex] - Slope, dimensionless.
[tex]b[/tex] - y-Intercept, dimensionless.
If we replace all variables with the components of the midpoints ([tex]A'' = (1,3)[/tex], [tex]C'' = (-2, -3)[/tex]), then we get a system of two linear equations:
[tex]m + b = 3[/tex] (Eq. 1)
[tex]-2\cdot m + b = -3[/tex] (Eq. 2)
Lastly, we proceed to solve the system algebraically:
In (Eq. 1):
[tex]b = 3-m[/tex]
(Eq. 1) in (Eq. 2):
[tex]-2\cdot m +3-m = -3[/tex]
[tex]-3\cdot m + 3 = -3[/tex]
[tex]-3\cdot m = -6[/tex]
[tex]m = 2[/tex]
By (Eq. 1):
[tex]b = 3-2[/tex]
[tex]b = 1[/tex]
The equation of the line of reflection is [tex]y = 2\cdot x +1[/tex].
