https://file.aiquickdraw.com/mathbot/file/1726173977892bm4mdl78.jpg

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 \( x = -1 \).

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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.

Graph the image of ΔKLM after a reflection over the line x = -1.

Learn how to reflect triangle ΔKLM over the vertical line x = -1. This geometry problem explains the reflection process with coordinates and includes step-by-step calculations. Suitable for students in Grades 7-10.

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