The point A(3,4) is reflected in the line x=2 and then reflected in the line y=1. Find the coordinates of the imageof A under these two reflections?
The point A(3,4) is reflected in the line x=2 and then reflected in the line y=1.
1. Find the coordinates of the image of A under these two reflections?
2. State the coordinatesof the point which remains invariant under these two reflections?
Detailed solution please.
To find the coordinates of the image of point A(3,4) after being reflected in the line x=2 and then reflected in the line y=1, let's go step by step:
First, let's reflect point A(3,4) in the line x=2.
Step 1: Reflecting in x=2
When reflecting a point in a vertical line, you switch the x-coordinate while keeping the y-coordinate the same.
Given A(3,4), to reflect in the line x=2, we flip the x-coordinate to the other side of the line, which is 2 units to the left of x=2.
The new x-coordinate becomes (2 - (3-2)) = (2 - 1) = 1.
The y-coordinate remains the same at 4.
So, after reflecting in the line x=2, the point A(3,4) will become A'(1,4).
Step 2: Reflecting in y=1
Now, to reflect point A'(1,4) in the line y=1, we switch the y-coordinate while keeping the x-coordinate the same.
Given A'(1,4), to reflect in the line y=1, we flip the y-coordinate to the other side of the line, which is 2 units below y=1.
The x-coordinate remains the same at 1.
The new y-coordinate becomes (1 - (4-1)) = (1 - 3) = -2.
So, after reflecting in the line y=1, the point A'(1,4) will become A''(1,-2).
Therefore, the coordinates of the image of point A(3,4) after being reflected in the line x=2 and then reflected in the line y=1 are A''(1,-2).