Analyzing Compositions of Dilations
The composition DO,0.75(x,y) ∘ DO,2(x,y) is applied to △LMN to create △L''M''N''.

On a coordinate plane, triangle L M N has points (negative 1, 2), (negative 1, negative 1), and (2, negative 1).

Which statements must be true regarding the two triangles? Check all that apply.

∠M ≅ ∠M''
△LMN ~ △L''M''N''
△LMN ≅ △L''M''N''
The coordinates of vertex L'' are (-3, 1.5).
The coordinates of vertex N'' are (3, -1.5).
The coordinates of vertex M'' are (1.5, -1.5).

We are given the triangle \( \triangle LMN \) with vertices at \( L(-1, 2) \), \( M(-1, -1) \), and \( N(2, -1) \), and a sequence of two dilations: \( DO, 2(x, y) \) followed by \( DO, 0.75(x, y) \).

### Steps to analyze the problem:

1. **Dilation Transformation**:
   - **First dilation \( DO, 2(x, y) \)**: This dilation scales the coordinates by a factor of 2. This means that the coordinates of each vertex will be doubled in distance from the origin.
   
     - For point \( L(-1, 2) \), after dilation by a factor of 2, the new coordinates are \( (-2, 4) \).
     - For point \( M(-1, -1) \), after dilation by a factor of 2, the new coordinates are \( (-2, -2) \).
     - For point \( N(2, -1) \), after dilation by a factor of 2, the new coordinates are \( (4, -2) \).

2. **Second dilation \( DO, 0.75(x, y) \)**: This dilation scales the coordinates by a factor of 0.75. Each point will be scaled by this factor from the origin.
   
   - For point \( L(-2, 4) \), after dilation by 0.75, the new coordinates are \( (-1.5, 3) \).
   - For point \( M(-2, -2) \), after dilation by 0.75, the new coordinates are \( (-1.5, -1.5) \).
   - For point \( N(4, -2) \), after dilation by 0.75, the new coordinates are \( (3, -1.5) \).

### Now, let's analyze each statement:

1. **∠M ≅ ∠M''**:  
   Since dilations preserve angles, the angle at \( M \) in \( \triangle LMN \) will be congruent to the angle at \( M'' \) in \( \triangle L''M''N'' \).  
   **This statement is true.**

2. **△LMN ~ △L''M''N''**:  
   Dilations produce similar triangles because they preserve angles and the sides are proportional.  
   **This statement is true.**

3. **△LMN ≅ △L''M''N''**:  
   Dilations do not preserve the actual size of the triangle, only its shape. Therefore, \( \triangle LMN \) is not congruent to \( \triangle L''M''N'' \) (it is only similar).  
   **This statement is false.**

4. **The coordinates of vertex L'' are (-3, 1.5)**:  
   From our calculations, the coordinates of \( L'' \) are \( (-1.5, 3) \), not \( (-3, 1.5) \).  
   **This statement is false.**

5. **The coordinates of vertex N'' are (3, -1.5)**:  
   From our calculations, the coordinates of \( N'' \) are indeed \( (3, -1.5) \).  
   **This statement is true.**

6. **The coordinates of vertex M'' are (1.5, -1.5)**:  
   From our calculations, the coordinates of \( M'' \) are \( (-1.5, -1.5) \), not \( (1.5, -1.5) \).  
   **This statement is false.**

### Final Answers:
- The following statements are true:
  - \( \angle M \cong \angle M'' \)
  - \( \triangle LMN \sim \triangle L''M''N'')
  - The coordinates of vertex N'' are \( (3, -1.5) \).

Would you like more details on the transformations, or do you have any further questions? 

### Expanding Questions:
1. What are some properties of dilations that help preserve similarity?
2. How do dilations affect the side lengths of triangles?
3. Can dilations be used to find the center of a triangle?
4. How would a dilation with a factor of 0.5 affect a triangle's size and coordinates?
5. What other transformations preserve congruence or similarity of triangles?

### Tip:
When working with dilations, remember that the center of dilation (in this case, the origin) is key in determining how the coordinates of the vertices are scaled.

The composition DO,0.75(x,y) ∘ DO,2(x,y) is applied to △LMN to create △L''M''N''. On a coordinate plane, triangle L M N has points (negative 1, 2), (negative 1, negative 1), and (2, negative 1). Which statements must be true regarding the two triangles? Check all that apply. ∠M ≅ ∠M'' △LMN ~ △L''M''N'' △LMN ≅ △L''M''N'' The coordinates of vertex L'' are (-3, 1.5). The coordinates of vertex N'' are (3, -1.5). The coordinates of vertex M'' are (1.5, -1.5).

Learn how to analyze the compositions of two dilations applied to triangle LMN on a coordinate plane. The solution explains how to determine congruent angles, similar triangles, and the new coordinates of vertices after the dilations. Suitable for students in Grades 8-10.

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