If point P(4,11) is reflected across the line y = 3 what are the coordinates of its reflection image?
(1 point)
A. (2, 11)
B.(11, 4)
C. (4, - 11)
D. (4, - 5)
To reflect a point across a line, we need to imagine a perpendicular line passing through the given point P and intersecting the given line y = 3. The reflection image will then be the point formed by the intersection of the perpendicular line and the given line.
Since the given line is y = 3, the y-coordinate of the reflection image will remain the same. Therefore, the y-coordinate of the reflection image will be 3.
To find the x-coordinate of the reflection image, we need to find the x-coordinate of the intersection point. We can find the equation of the perpendicular line passing through point P(4,11) by finding its slope, which is the negative reciprocal of the slope of the given line.
The slope of the given line y = 3 is 0, so the slope of the perpendicular line will be undefined (or infinite). Since the line is perpendicular to the given line, it will be vertical.
The equation of a vertical line passing through (4,11) is x = 4.
Now, we need to find the intersection point of the line y = 3 and the line x = 4. Since the y-coordinate of the intersection point is 3, and the x-coordinate is 4, the coordinates of the reflection image will be (4, 3).
Therefore, the coordinates of the reflection image are:
A. (2, 11)
B. (11, 4)
C. (4, -11)
D. (4, -5)
The correct answer is D. (4, -5).
To find the reflection image of point P(4,11) across the line y = 3, we need to find the new coordinates of the point after reflection.
Step 1: Find the distance between the line y = 3 and the y-coordinate of point P.
Distance = 11 - 3 = 8
Step 2: Reflect the point across the line y = 3 with the same distance.
The y-coordinate of the reflection image will be 3 - 8 = -5.
Step 3: The x-coordinate of the reflection image remains the same.
The x-coordinate of the reflection image is 4.
Therefore, the coordinates of the reflection image are (4, -5).
So, the correct answer is D. (4, -5).