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Answer:
[tex]y = x[/tex]
Step-by-step explanation:
Given
[tex]A =(6,3.7)[/tex]
[tex]B = (5.4,2)[/tex]
[tex]C = (1,3)[/tex]
[tex]A' = (3.7, 6)[/tex]
[tex]B' = (2, 5.4)[/tex]
[tex]C' = (3, 1)[/tex]
Required
The equation for the line of reflection
Taking points A and A' as a point of reference:
[tex]A =(6,3.7)[/tex] and [tex]A' = (3.7, 6)[/tex]
First, calculate the midpoint (M)
[tex]M = (\frac{x_2+x_1}{2},\frac{y_2+y_1}{2})[/tex]
[tex]M = (\frac{3.7+6}{2},\frac{6+3.7}{2})[/tex]
[tex]M = (\frac{9.7}{2},\frac{9.7}{2})[/tex]
[tex]M = (4.85, 4.85)[/tex]
Calculate the slope (m)
[tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]
[tex]m = \frac{6 - 3.7}{3.7 - 6}[/tex]
[tex]m = \frac{2.3}{-2.3}[/tex]
[tex]m = -1[/tex]
The midpoint of a reflection is always perpendicular to the points being reflected.
So, the slope of [tex]M = (4.85, 4.85)[/tex] is:
[tex]m_2 = -\frac{1}{m}[/tex]
[tex]m_2 = -\frac{1}{-1}[/tex]
[tex]m_2 = 1[/tex]
The equation is then calculated as:
[tex]y = m_2(x - x_1) + y_1[/tex]
Where:
[tex](x_1,y_1) = (4.85,4.85)[/tex]
[tex]y = 1(x - 4.85) + 4.85[/tex]
[tex]y = x - 4.85 + 4.85[/tex]
[tex]y = x[/tex]
Hence, the equation of line of reflection is:
[tex]y = x[/tex]