Write the equation of the mirror line for the segment with endpoints (5, 4) and (-1, -1).
find the reflection of point a(3, -2) through the mirror line of y = 1/2x − 1.

  a) 12x +10y = 39

  b) A'(1, 2)

Step-by-step explanation:

a) The mirror line is the perpendicular bisector of the segment between the given points. The difference in their coordinates is ...

  (Δx, Δy) = (5, 4) -(-1, -1) = (6, 5)

The midpoint between the given points is ...

  ((5, 4) +(-1, -1))/2 = (2, 3/2)

One way to write the equation of the perpendicular line through point (h, k) is ...

  Δx(x -h) +Δy(y -k) = 0

Filling in the numbers from above, this becomes ...

  6(x -2) +5(y -3/2) = 0

  6x +5y -39/2 = 0 . . . . . . eliminate parentheses

  12x +10y = 39 . . . . . . . . . mirror line in standard form

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b) The line through A perpendicular to the mirror line will have a slope that is the negative reciprocal of that of the mirror line: -1/(1/2) = -2. Then the point-slope equation of the line through A is ...

  y +2 = -2(x -3)

Using the equation of the mirror line to substitute for y, we can find the point on the mirror line that is the midpoint between A and its reflection.

  (1/2x -1) +2 = -2(x -3)

  1/2x +1 = -2x +6 . . . . . . eliminate parentheses

  5/2x = 5 . . . . . . . . . . . . add 2x-1

  x = 2 . . . . . . . . . . . . . . . multiply by 2/5

  y = (1/2)(2) -1 = 0 . . . . . find the y-coordinate of the midpoint using the equation for y

Now we know that (x, y) = (2, 0) is the midpoint between A and A'.

  (2, 0) = (A +A')/2 . . . . equation for midpoint of A and A'

  (4, 0) = A +A' . . . . . . . multiply by 2

  (4, 0) - A = A' = (4, 0) -(3, -2) . . . . subtract A

  A' = (1, 2) . . . . . the reflection of point A