https://file.aiquickdraw.com/mathbot/file/1729271266690iezeotqt.jpg

The image shows a triangle on a coordinate plane with vertices labeled \( K \), \( L \), and \( M \). Additionally, there are instructions and transformations applied to the triangle.

Let's break it down step by step:

### 1. Coordinates of the Vertices:
- Point \( K \) is located at \( (-2, 3) \).
- Point \( L \) is at \( (0, 0) \).
- Point \( M \) is at \( (4, 0) \).

### 2. Transformation:
There appears to be a transformation applied to the triangle, indicated by the notation \( T_{(x, y)} \). In the image, the transformation is given as:
\[
T_{(x=-1, y=2)}: \triangle KLM \rightarrow \triangle K'L'M'
\]
This means the triangle is translated by moving each point \( x \) units to the left and \( y \) units up. Specifically:
- \( x \)-coordinate is decreased by 1.
- \( y \)-coordinate is increased by 2.

### 3. Applying the Transformation:
Let's apply the transformation to each vertex:
- \( K(-2, 3) \rightarrow K'(-2-1, 3+2) = (-3, 5) \)
- \( L(0, 0) \rightarrow L'(0-1, 0+2) = (-1, 2) \)
- \( M(4, 0) \rightarrow M'(4-1, 0+2) = (3, 2) \)

The new triangle \( \triangle K'L'M' \) has the following coordinates:
- \( K'(-3, 5) \)
- \( L'(-1, 2) \)
- \( M'(3, 2) \)

### Conclusion:
The transformation results in a translation of the triangle. The new coordinates are as given above.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How would you calculate the area of \( \triangle KLM \) before and after the transformation?
2. Can you describe how a reflection transformation would change the triangle?
3. How does a rotation of the triangle \( \triangle KLM \) by 90° around the origin affect the coordinates?
4. How do you find the slope of the line connecting points \( K \) and \( M \)?
5. Can you determine the perimeter of the triangle before and after translation?

**Tip:** For translation transformations, always remember to adjust both \( x \) and \( y \)-coordinates by the given values simultaneously to avoid mistakes.

Translate triangle KLM given vertices K(-2, 3), L(0, 0), and M(4, 0) using the transformation T_{(x=-1, y=2)}.

Explore the translation of triangle KLM with vertices K(-2, 3), L(0, 0), and M(4, 0) using transformation T_{(x=-1, y=2)}. Learn how each point shifts and discover the new coordinates after the translation.

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