A(- x - 8) A( coordinates of ? gde reflection where the translation is (x, y) -> (x + 5, y) , and the line of reflection is y = 1 What are the
(1 point)
A. (2 - 3)
B. (- 3.5)
C. (5, 0)
D. (2.5)
To find the reflection of point A(- x - 8) across the line y = 1, we need to find the distance between the point and the line and then apply the translation (x, y) -> (x + 5, y).
The distance between a point (x, y) and the line y = 1 is given by the equation: distance = |y - 1|
For point A(- x - 8), the distance would be: distance = |- x - 8 - 1| = |- x - 9|
Now, we need to reflect point A across the line y = 1 by taking the same distance on the opposite side of the line. So, the reflected point would be (- x - 8, 1 + |- x - 9|).
Applying the translation (x, y) -> (x + 5, y) to the reflected point, we get the following coordinates:
(- x - 8) -> (- x - 8 + 5) = (- x - 3)
(1 + |- x - 9|) -> (1 + |- x - 9|)
Therefore, the coordinates of the reflected point would be (- x - 3, 1 + |- x - 9|).
Based on the options given, none of them match the coordinates of the reflected point.
To find the coordinates of the reflection of point A(-x, -8) across the line y = 1 with the translation (x, y) -> (x + 5, y), we can follow these steps:
Step 1: Apply the translation by adding 5 to the x-coordinate: A(-x + 5, -8).
Step 2: Find the distance between the original point A and the line of reflection y = 1. The perpendicular distance between a point (x1, y1) and a line ax + by + c = 0 can be calculated using the formula:
distance = | ax1 + by1 + c | / sqrt(a^2 + b^2)
In this case, the equation of the line y = 1 can be rewritten as 0x + 1y - 1 = 0. The coordinates of A are (-x + 5, -8), and substituting these values into the formula, we get:
distance = | 0(-x + 5) + 1(-8) - 1 | / sqrt(0^2 + 1^2)
distance = | -x + 5 - 8 - 1 | / 1
distance = | -x - 4 | / 1
distance = |-x - 4|
Step 3: Since the line of reflection is y = 1, the image point will have the same y-coordinate. Therefore, the y-coordinate of the image point is 1.
Step 4: For the image point, the distance from the line of reflection will be the same as the distance from the original point A. So, we can compare the distance |-x - 4| with the distance from the line y = 1.
Step 5: If |-x - 4| is a positive number, the x-coordinate of the image point will be negativex - 4. If |-x - 4| is negative, the x-coordinate of the image point will be - (x - 4).
Using this information, we can determine the image point's coordinates:
- If |-x - 4| > 0, the image point is (- (x - 4), 1).
- If |-x - 4| = 0, the image point is (4 + x, 1).
Therefore, the correct option would be D. (2.5), as none of the other options match the coordinates we have derived using the formula.