To address the given problem, we start with the transformation [tex]\( T : (x, y) \rightarrow (x+2, y+1) \)[/tex], applied to two points in the plane.
1. Given Transformation:
The transformation [tex]\( T \)[/tex] is defined such that any point [tex]\((x, y)\)[/tex] is mapped to [tex]\((x+2, y+1)\)[/tex].
2. Identify Points A and B:
Suppose we are working with points [tex]\(A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex].
- Here, the given points are not explicitly stated, so let's assume they are denoted by A and B.
3. Apply the Transformation:
Using the transformation [tex]\( T \)[/tex]:
- The new coordinates of point [tex]\( A \)[/tex] are [tex]\((x_1', y_1') = (x_1 + 2, y_1 + 1)\)[/tex]
- The new coordinates of point [tex]\( B \)[/tex] are [tex]\((x_2', y_2') = (x_2 + 2, y_2 + 1)\)[/tex]
4. Concrete Calculation:
Let's assume the original coordinates:
- Suppose point [tex]\( A \)[/tex]: [tex]\((x_1 = 6, y_1 = 3)\)[/tex]
- Suppose point [tex]\( B \)[/tex]: [tex]\((x_2 = 6, y_2 = 5)\)[/tex]
Apply the transformation:
- For point [tex]\( A \)[/tex]: [tex]\((x_1', y_1') = (6 + 2, 3 + 1) = (8, 4)\)[/tex]
- For point [tex]\( B \)[/tex]: [tex]\((x_2', y_2') = (6 + 2, 5 + 1) = (8, 6)\)[/tex]
5. Calculate the Distance [tex]\( A'B' \)[/tex]:
Using the distance formula:
[tex]\[
d = \sqrt{(x_2' - x_1')^2 + (y_2' - y_1')^2}
\][/tex]
Plugging in the transformed coordinates:
[tex]\[
d = \sqrt{(8 - 8)^2 + (6 - 4)^2} = \sqrt{0^2 + 2^2} = \sqrt{4} = 2
\][/tex]
Thus, the distance [tex]\( A'B' \)[/tex] after the transformation is:
[tex]\[
\boxed{2.0}
\][/tex]
