To reflect the triangle $\Delta KLM$ over the line $x = -1$, we need to reflect each vertex of the triangle (points $K$, $L$, and $M$) over this vertical line.

The reflection process for any point $(x, y)$ over a vertical line $x = c$ can be computed by keeping the $y$-coordinate the same and adjusting the $x$-coordinate to be symmetrically reflected over the line $x = c$. Specifically, the new $x'$-coordinate will be calculated as: $x' = 2c - x$ where $c = -1$ in this case.

Step-by-step reflection:

1. Coordinates of point $K$:

Current coordinates: $K(4, 2)$

Reflection over $x = -1$: $x' = 2(-1) - 4 = -2 - 4 = -6$ New coordinates: $K'(-6, 2)$

2. Coordinates of point $L$:

Current coordinates: $L(6, 0)$

Reflection over $x = -1$: $x' = 2(-1) - 6 = -2 - 6 = -8$ New coordinates: $L'(-8, 0)$

3. Coordinates of point $M$:

Current coordinates: $M(2, 6)$

Reflection over $x = -1$: $x' = 2(-1) - 2 = -2 - 2 = -4$ New coordinates: $M'(-4, 6)$

Final coordinates of the reflected triangle:

• $K'(-6, 2)$
• $L'(-8, 0)$
• $M'(-4, 6)$

You can now plot these points to reflect the triangle over the line $x = -1$.

Would you like further details on how to graph it or have any other questions? Here are five related questions for further exploration:

1. How would the reflection change if the line was $x = 2$ instead of $x = -1$?
2. How do reflections over the $y$-axis differ from reflections over other vertical lines?
3. What is the formula for reflecting a point over a line such as $y = c$?
4. How would this problem differ if the line of reflection were diagonal, like $y = x$?
5. How can you verify that the reflected triangle is congruent to the original?

Tip: Always double-check the sign and magnitude of the reflection when working with vertical or horizontal lines.