If AB = 27 inches, then find the distance of A’B’ after a reflection over the x-axis and a translation of (x+6,y-2)
To find the distance of A'B' after a reflection over the x-axis and a translation of (x + 6, y - 2), we need to apply these transformations to point A and point B, and then calculate the distance between the transformed points A' and B'.
Let's start with the reflection over the x-axis.
Reflection over the x-axis means that the y-coordinate will change sign while the x-coordinate remains the same. This results in point A becoming A' with coordinates (x, -y) and point B becoming B' with coordinates (x, -y).
Next, let's apply the translation of (x + 6, y - 2) to A' and B'.
For point A': (x, -y) + (x + 6, y - 2) = (2x + 6, -2)
For point B': (x, -y) + (x + 6, y - 2) = (2x + 6, -4y)
Now, let's calculate the distance between A' and B' using the distance formula:
Distance = sqrt((2x + 6 - 2x + 6)^2 + (-4y - (-2))^2)
Distance = sqrt(12^2 + (-4y + 2)^2)
Distance = sqrt(144 + (-4y + 2)^2)
Distance = sqrt(144 + 16y^2 - 16y + 4)
Distance = sqrt(16y^2 - 16y + 148)
Therefore, the distance of A'B' after the reflection over the x-axis and a translation of (x + 6, y - 2) is sqrt(16y^2 - 16y + 148) inches.