What is the image of A(4,4) under a reflection in the point P(4,-2)?
To find the image of point A(4,4) under a reflection in point P(4,-2), we can follow these steps:
1. Draw a line passing through point A(4,4) and perpendicular to the line connecting A and P. This line will serve as the perpendicular bisector of the segment joining A and its reflection.
2. Find the midpoint M of the segment joining A and P. The midpoint is calculated by taking the average of the x-coordinates and the average of the y-coordinates of A and P.
- For the x-coordinate: (4 + 4) / 2 = 8 / 2 = 4
- For the y-coordinate: (4 + (-2)) / 2 = 2 / 2 = 1
Hence, the midpoint M is at coordinates (4,1).
3. Extend the line from step 1 through point M until it intersects with the line connecting A and P. This point of intersection will be the image of A under the reflection.
Since point A is equidistant from both P and its image, the image will lie on the extended line passing through M.
4. Since the line connecting A and P is vertical, the reflection will occur horizontally. Therefore, the x-coordinate of the image will be the same as the x-coordinate of point M.
Thus, the image of A(4,4) under a reflection in P(4,-2) will have an x-coordinate of 4.
5. To find the y-coordinate of the image, substitute the x-coordinate, 4, into the equation of the line connecting A and P:
(x - 4) / (4 - 4) = (y - 4) / (-2 - 4)
Simplifying the equation:
(x - 4) / 0 = (y - 4) / (-6)
This equation is undefined since division by zero is not defined.
Therefore, the y-coordinate of the image is undefined.
Thus, the image of A(4,4) under a reflection in P(4,-2) is a point with coordinates (4, undefined).