The image of a point after a reflection over the line y = –x is (7, –1). Find the coordinates of the preimage.
ummh, doesn't it say the y is the opposite of the x ?
(and the x is the opposite of the y ?)
so (7,-1) ----> (1, -7)
check:
plot the points and see how they relate to the line y = -x
To find the coordinates of the preimage, we need to understand the concept of reflection over a line.
A reflection over a line is a transformation that flips a point across a given line. In this case, the given line is y = -x.
When a point is reflected over a line, the distance from the point to the line before and after the reflection is the same, but the direction is reversed.
Let's denote the original point as (x, y) and the reflected point as (x', y'). In this case, we know that the reflected point is (7, -1).
To find the preimage, we need to undo the reflection. Since the line of reflection is y = -x, we need to find the line perpendicular to this line that passes through the reflected point (7, -1).
The perpendicular line to y = -x will have a slope of the negative reciprocal of the slope of the given line. The given line has a slope of -1, so the perpendicular line will have a slope of 1.
Using the point-slope form of a line, we can write the equation for the perpendicular line passing through point (7, -1) as:
y - (-1) = 1(x - 7)
Simplifying the equation:
y + 1 = x - 7
y = x - 8
Now we have the equation for the line through the preimage and reflected point.
To find the preimage coordinates, we need to find the intersection point of this line and the line of reflection, y = -x.
Substitute y = -x in the equation y = x - 8:
-x = x - 8
2x = 8
x = 4
Substitute x = 4 back into the equation y = x - 8:
y = 4 - 8
y = -4
Therefore, the coordinates of the preimage are (4, -4).