In a reflection the image of the line y-2x=3 is the line 2y-x=9. Find the axis of reflection.
The axis of reflection is the perpendicular bisector of the segment connecting any point on the original line to its corresponding point on the reflected line.
To find a convenient point on the original line, let's first rewrite the equation of the line y-2x=3 in slope-intercept form:
y = 2x + 3
One point on this line is (0, 3). Its corresponding point on the reflected line is the intersection of the two lines. We can find this point by solving the system of equations:
y - 2x = 3
2y - x = 9
Multiplying the first equation by 2 and adding it to the second equation, we get:
5y = 15
So y = 3. Substituting into the first equation, we get:
3 - 2x = 3
So x = 0.
Therefore, the corresponding point is (0, 3).
Now we can find the midpoint of the segment connecting (0, 3) to its reflection across the axis of reflection (let's call it (a, b)). The midpoint of this segment lies on the axis of reflection.
The midpoint is:
((0 + a)/2, (3 + b)/2)
And its reflection across the axis of reflection is:
((a + 2b - 18)/5, (2a + b - 3)/5)
Setting these two points equal (since they are reflections across the axis of reflection), we get the system:
(a + 2b - 18)/5 = (0 + a)/2
(2a + b - 3)/5 = (3 + b)/2
Solving this system, we get:
a = 12
b = 1
Therefore, the axis of reflection is the line passing through the point (12, 1) and perpendicular to the original line y-2x=3.
The original line has slope 2, so the axis of reflection has slope -1/2 (since it is perpendicular).
Using point-slope form with the point (12, 1), we get the equation of the axis of reflection:
y - 1 = (-1/2)(x - 12)
Simplifying, we get:
2y - x = 14
Therefore, the axis of reflection is the line 2y - x = 14.
To find the axis of reflection, we need to determine the line that serves as the mirror for the given reflection. The axis of reflection is defined as the line that is equidistant from each point on the original line and its corresponding image.
To determine the axis of reflection, let's consider two points on the original line y - 2x = 3. We can choose any two convenient points.
Let's choose x = 0, then y = 3. This gives us the point (0, 3) on the original line.
Let's choose x = 1, then y = 5. This gives us the point (1, 5) on the original line.
Now, let's find the corresponding points on the reflected line 2y - x = 9.
For the point (0, 3), substituting these values into 2y - x = 9, we get:
2(3) - 0 = 9
6 = 9
Since this equation is not true, the reflected point for (0, 3) is not on the reflected line.
Similarly, for the point (1, 5), substituting these values into 2y - x = 9, we get:
2(5) - 1 = 9
9 = 9
Since this equation is true, the reflected point for (1, 5) is on the reflected line.
Therefore, we can conclude that the line 2y - x = 9 is the mirror for the reflection of the line y - 2x = 3.
To find the axis of reflection, we can find the equation of the line that is perpendicular to the mirror line and passes through its midpoint. Let's find the midpoint of the line (0, 3) and (1, 5).
Midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint = ((0 + 1)/2, (3 + 5)/2)
Midpoint = (1/2, 8/2)
Midpoint = (1/2, 4)
The midpoint of the line is (1/2, 4).
To find the slope of the mirror line 2y - x = 9, we need to rearrange it into slope-intercept form (y = mx + b), where m is the slope:
2y - x = 9
2y = x + 9
y = (1/2)x + 9/2
The slope of the mirror line is 1/2.
To find the slope of the line perpendicular to the mirror line, we can take the negative reciprocal of the slope of the mirror line.
The negative reciprocal of 1/2 is -2.
We can use the point-slope form of a linear equation to find the equation of the line that passes through the midpoint (1/2, 4) with a slope of -2.
Point-slope form:
y - y1 = m(x - x1)
Using the midpoint (1/2, 4) and slope -2:
y - 4 = -2(x - 1/2)
Simplifying:
y - 4 = -2x + 1
y = -2x + 5
Therefore, the equation of the line that serves as the axis of reflection is y = -2x + 5.