What is the equation of the image of the line L:y=x+3 after reflection about the line y = x-2?
a few seconds with google yielded
https://math.stackexchange.com/questions/67024/the-equation-of-a-line-reflected-about-another-line
I will do
To find the equation of the image of the line L: y = x + 3 after reflection about the line y = x - 2, we can follow the steps below:
Step 1: Find the slope and y-intercept of line L.
The given equation of line L is y = x + 3.
Therefore, the slope of line L is 1 and the y-intercept is 3.
Step 2: Find the slope of the line of reflection.
The given equation of the line of reflection is y = x - 2.
The slope of the line of reflection is 1.
Step 3: Find the slope of the line perpendicular to the line of reflection.
Since the line of reflection has a slope of 1, the slope of the line perpendicular to it is -1 (negative reciprocal).
Step 4: Find the equation of the line L' perpendicular to the line of reflection and passing through the intersection of line L and the line of reflection.
The slope of L' is -1.
To find the y-intercept of L', we need to find the point where line L intersects the line of reflection.
Setting the equations of L and the line of reflection equal to each other, we get:
x + 3 = x - 2
Simplifying, we find that x = 2.
Substituting x = 2 back into the equation of L, we find that y = 5.
Therefore, the point of intersection is (2, 5).
Using the point-slope form of a linear equation (y - y1 = m(x - x1)), we can write the equation of L' as:
y - 5 = -1(x - 2)
Simplifying, we find that y - 5 = -x + 2
Rearranging, the equation of L' is y = -x + 7.
Step 5: Find the equation of the image of line L after reflection about the line of reflection.
Since the image of a line after reflection is the line itself, but with the sign of its slope negated, the equation of the image of L is:
y = -x + 7.
Therefore, the equation of the image of the line L: y = x + 3 after reflection about the line y = x - 2 is y = -x + 7.
To find the equation of the image of a line after reflection about another line, we can follow these steps:
1. Find the equation of the line of reflection: In this case, the line of reflection is y = x - 2.
2. Find the slope of the line of reflection: The slope of the line y = x - 2 is 1.
3. Find the equation of the perpendicular line: The image of a point reflected over a given line lies on the perpendicular line drawn from that point to the line of reflection.
- Since the slope of the line of reflection is 1, the slope of the perpendicular line is -1 (negative reciprocal).
- The midpoint of the line L:y=x+3 will lie on the line of reflection, so we can find two points on the line L to determine the equation of the perpendicular line.
Let's choose two points: P(0, 3) and Q(1, 4) on the line L.
4. Determine the midpoint of the line segment PQ: The midpoint is given by the average of the x-coordinates and the average of the y-coordinates.
- Midpoint of PQ: [(0 + 1) / 2, (3 + 4) / 2] = (1/2, 7/2)
5. Find the equation of the perpendicular line using the midpoint and the slope: We can use the point-slope form of a line to find the equation.
Point-slope form: y - y1 = m(x - x1), where (x1, y1) is the midpoint and m is the slope.
- Using the midpoint (1/2, 7/2) and the slope -1, we get: y - (7/2) = -1(x - 1/2)
- Simplifying the equation: y - (7/2) = -x + 1/2
- Rearranging the equation in the slope-intercept form (y = mx + b): y = -x + 9/2
Therefore, the equation of the image of the line L:y = x + 3 after reflection about the line y = x - 2 is y = -x + 9/2.