To find the coordinates of $K'$ and $M'$, we need to apply the translation vector $\left( -6, 2 \right)$ to the original coordinates of points $K$ and $M$.

Step 1: Identify the original coordinates

From the graph:

• Point $K$ is located at $(4, 1)$.
• Point $M$ is located at $(1, -5)$.

Step 2: Apply the translation vector

The translation vector is $\left( -6, 2 \right)$, which means we subtract 6 from the $x$-coordinate and add 2 to the $y$-coordinate.

a) Finding the coordinates of $K'$:

• Original coordinates of $K$: $(4, 1)$.
• Applying the translation: $K' = (4 - 6, 1 + 2) = (-2, 3).$

b) Finding the coordinates of $M'$:

• Original coordinates of $M$: $(1, -5)$.
• Applying the translation: $M' = (1 - 6, -5 + 2) = (-5, -3).$

Final Answers:

• The coordinates of $K'$ are $(-2, 3)$.
• The coordinates of $M'$ are $(-5, -3)$.

Would you like further details or have any questions?

Related Questions:

1. What would be the new coordinates if the translation vector were $(3, -4)$?
2. How would the shape of the figure change if the vector had a magnitude of $\sqrt{45}$ in a different direction?
3. How can we determine the translation vector given the original and translated coordinates of any two points?
4. What is the effect of translating a shape by a vector with both positive components?
5. How do we find the midpoint of a line segment before and after translation?

Tip:

Always double-check the direction of translation by noting if the vector components are positive or negative, which indicates movement along the respective axes.