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To solve this problem, we need to reflect the given points \( L \), \( M \), \( K \), and \( N \) over the line \( y = -x \). Here's a step-by-step explanation:

### Step 1: Identify the original coordinates
Let's find the original coordinates of each vertex:
- \( L = (2, 9) \)
- \( M = (5, 9) \)
- \( K = (2, 6) \)
- \( N = (5, 6) \)

### Step 2: Reflect the points over the line \( y = -x \)
When reflecting a point over the line \( y = -x \), the transformation rule is to swap the x and y coordinates and then change their signs. That is, a point \( (a, b) \) reflects to \( (-b, -a) \).

Applying this rule to each point:
- For \( L = (2, 9) \), the reflected point is \( L' = (-9, -2) \)
- For \( M = (5, 9) \), the reflected point is \( M' = (-9, -5) \)
- For \( K = (2, 6) \), the reflected point is \( K' = (-6, -2) \)
- For \( N = (5, 6) \), the reflected point is \( N' = (-6, -5) \)

### Step 3: The reflected coordinates
The vertices after reflection over the line \( y = -x \) are:
- \( L' = (-9, -2) \)
- \( M' = (-9, -5) \)
- \( K' = (-6, -2) \)
- \( N' = (-6, -5) \)

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the general rule for reflecting a point over the line \( y = x \)?
2. How would you reflect a figure over the x-axis?
3. What happens to the coordinates when a point is reflected over the y-axis?
4. How do you determine the line of reflection between two points?
5. How would you describe the reflection of a figure over any arbitrary line, not just \( y = -x \)?

**Tip:** Always remember that reflecting over a line involves swapping coordinates and changing their signs as per the transformation rules associated with that line.

Write the coordinates of the vertices after a reflection over the line y = -x.

Solve a geometry problem involving reflections of points over the line y = -x. Learn how to reflect coordinates using the transformation rule (x, y) → (-y, -x). Suitable for Grades 7-9 geometry students.

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