Question

In a transformation of triangle ABC to its image A' B' C', the coordinates of A are (-2, 6) and the coordinates of A' are (2, 2). What is the equation of the line of reflection that relates the original and image triangles?

A) y = -x + 4
B) y = x - 4
C) y = -x - 4
D) y = x + 4

Answer 1

The equation of the line of reflection that relates triangle ABC to its image A' B' C' is y = x + 4.

Step-by-step explanation:

The equation of the line of reflection between two points (A and A') can be found by determining the equation of the perpendicular bisector of the segment connecting the two points. The coordinates of point A are (-2, 6), and the coordinates of A' are (2, 2). The midpoint M between these two points has coordinates (mx, my) where mx = (xA + xA')/2 and my = (yA + yA')/2. In this case, the midpoint M is (0, 4).

The slope of the line through A and A' is (yA' - yA) / (xA' - xA) which simplifies to (2 - 6) / (2 - (-2)) = -4/4 = -1. The slope of the line of reflection, which is perpendicular to this line, should be the negative reciprocal of -1, which is 1. Thus, the line of reflection has the slope 1 and passes through point M (0, 4).

Using point-slope form, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the midpoint, the equation of the line of reflection is y - 4 = 1(x - 0), which simplifies to y = x + 4, making D) the correct answer.