Question

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.

Answer 1

The equation of the mirror line in standard form is: 10y+12x=39. The reflection of the point is A'(1,2)

Step-by-step explanation:

Given that the endpoints of the segments are: (5,4) and (-1,-1) then the equation of the mirror line will be;

slope=m=Δy/Δx

Δy= -1 - 4= -5

Δx= -1-5 = -6

m= -5/-6 = 5/6

For the equation, taking point (5,4), and (x,y), it will be;

m=Δy/Δx

5/6= y-4/x-5

5(x-5) = 6(y-4)

5x-25 =6y-24

5x=6y-24+25

5x=6y+1

5x-1=6y

6y=5x-1 -------divide both sides by 6

y=5/6x -1/6 -------Equation of line segment

The mirror line is a perpendicular bisector of the segment.This means it passes at the midpoint of the segment.

Finding the midpoint of the segment will be;

{(x₁+x₂)/2 ,(y₁+y₂)/2}

{(5+-1)/2, (4+-1)/2} =(4/2, 3/2) =(2, 3/2)

Using the coordinate point of the midpoint, you can find the equation of the mirror line knowing that the slope will be -6/5 because the mirror line is perpendicular to the line segment that has a slope of 5/6.

Finding the equation of the mirror line using point (2, 3/2), (x,y) and m= -6/5

$\frac{y-1.5}{x-2} =\frac{-6}{5} \\\\5(y-1.5)=-6(x-2)\\5y-7.5=-6x+12\\5y-\frac{15}{2} =-6x+12\\\\10y-15=-12x+24\\\\10y+12x=15+24\\\\10y+12x=39$

The equation of the mirror line in standard form is;

10y+12x=39

2. Given point A(3,-2)  and line y=1/2x -1 ,you can use a graph tool to plot the equation for the mirror line, plot the object point and locate the image point across the mirror line.

From the graph, the image is A'(1,2)

Learn More

Reflection of a point :https://brainly.com/question/12865568

Keywords : equation, mirror line, segment, endpoints, reflection

#LearnwithBrainly

Answer 2

Answer:

  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

__

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