To find the coordinates of [tex]\( K' \)[/tex] after translating point [tex]\( K \)[/tex], we need to determine the translation vector that maps point [tex]\( J \)[/tex] to [tex]\( J' \)[/tex]. Here are the coordinates given:
- [tex]\( J(2, 4) \)[/tex]
- [tex]\( J'(3, 3) \)[/tex]
- [tex]\( K(3, 1) \)[/tex]
Step-by-Step Solution:
1. Determine the translation vector:
- The translation vector is calculated by finding the difference in the x-coordinates and y-coordinates of [tex]\( J \)[/tex] and [tex]\( J' \)[/tex]:
[tex]\[
\text{Translation vector} = (J'_x - J_x, J'_y - J_y)
\][/tex]
- Substitute the given coordinates of [tex]\( J \)[/tex] and [tex]\( J' \)[/tex]:
[tex]\[
\text{Translation vector} = (3 - 2, 3 - 4) = (1, -1)
\][/tex]
2. Apply the translation vector to point [tex]\( K \)[/tex]:
- To find the coordinates of [tex]\( K' \)[/tex], we add the components of the translation vector to the coordinates of [tex]\( K \)[/tex]:
[tex]\[
K'_x = K_x + 1
\][/tex]
[tex]\[
K'_y = K_y - 1
\][/tex]
- Substitute the given coordinates of [tex]\( K \)[/tex]:
[tex]\[
K'_x = 3 + 1 = 4
\][/tex]
[tex]\[
K'_y = 1 - 1 = 0
\][/tex]
3. Conclusion:
- Therefore, the coordinates of [tex]\( K' \)[/tex] are [tex]\((4, 0)\)[/tex].
So, the correct coordinates of [tex]\( K' \)[/tex] are [tex]\( \boxed{(4, 0)} \)[/tex].
