Answer :
Let's analyze the composition of dilations applied to triangle [tex]$\triangle LMN$[/tex] to create [tex]$\triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime}$[/tex] and determine which statements are true.
### Step-by-Step Solution:
1. Angles under dilation:
- Statement: [tex]$\angle M = \angle M^{\prime \prime}$[/tex].
- Reasoning: Dilation transformations preserve the angles of a shape. This means that the angles of [tex]$\triangle LMN$[/tex] will remain the same in [tex]$\triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime}$[/tex].
- Conclusion: This statement is true.
2. Similarity of triangles:
- Statement: [tex]$\triangle LMN \sim \triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime}$[/tex].
- Reasoning: A dilation transformation scales the sides of a triangle proportionally but does not change the shape of the triangle. This means the triangles are similar by definition.
- Conclusion: This statement is true.
3. Congruence of triangles:
- Statement: [tex]$\triangle LMN = \triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime}$[/tex].
- Reasoning: Congruence requires the triangles to have identical side lengths and identical angles. While the angles remain unchanged, dilation changes the side lengths by a scale factor. Since different scale factors (2 and 0.75) have been applied, the side lengths of the resultant triangle will differ from the original.
- Conclusion: This statement is false.
4. Coordinates of vertices:
Let's find the coordinates of the vertices after applying the given dilations.
- First dilation [tex]\(D_{O,2} (x,y)\)[/tex]:
- Scale factor is 2 (about the origin).
- The coordinates of a point [tex]\((x, y)\)[/tex] will become [tex]\((2x, 2y)\)[/tex].
- Second dilation [tex]\(D_{0,0.75} (x,y)\)[/tex]:
- Scale factor is 0.75 (about the origin).
- The coordinates of a point [tex]\((2x, 2y)\)[/tex] will be scaled down to [tex]\((0.75 \cdot 2x, 0.75 \cdot 2y) = (1.5x, 1.5y)\)[/tex].
The final coordinates of the vertices will be calculated based on their original coordinates.
- [tex]\(\mathbf{L}\)[/tex]:
- Original: [tex]\((-3, 1.5)\)[/tex]
- After [tex]\(D_{O, 2}\)[/tex]: [tex]\((-6, 3)\)[/tex]
- After [tex]\(D_{0, 0.75}\)[/tex]: [tex]\((-6 \cdot 0.75, 3 \cdot 0.75) = (-4.5, 2.25)\)[/tex]
- Given statement says the coordinates are [tex]\((-3, 1.5)\)[/tex].
- Conclusion: This statement is false.
- [tex]\(\mathbf{M}\)[/tex]:
- Original: [tex]\((1.5, -1.5)\)[/tex]
- After [tex]\(D_{O, 2}\)[/tex]: [tex]\((3, -3)\)[/tex]
- After [tex]\(D_{0, 0.75}\)[/tex]: [tex]\((3 \cdot 0.75, -3 \cdot 0.75) = (2.25, -2.25)\)[/tex]
- Given statement says the coordinates are [tex]\((1.5, -1.5)\)[/tex].
- Conclusion: This statement is false.
- [tex]\(\mathbf{N}\)[/tex]:
- Original: [tex]\((3, -1.5)\)[/tex]
- After [tex]\(D_{O, 2}\)[/tex]: [tex]\((6, -3)\)[/tex]
- After [tex]\(D_{0, 0.75}\)[/tex]: [tex]\((6 \cdot 0.75, -3 \cdot 0.75) = (4.5, -2.25)\)[/tex]
- Given statement says the coordinates are [tex]\((3, -1.5)\)[/tex].
- Conclusion: This statement is false.
### Summary:
The true statements regarding the dilations applied to [tex]\(\triangle LMN\)[/tex] are:
- [tex]$\angle M = \angle M^{\prime \prime}$[/tex]
- [tex]$\triangle LMN \sim \triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime}$[/tex]
Hence, the final selection will be:
- [tex]$\boxed{\angle M = \angle M^{\prime \prime}}$[/tex]
- [tex]$\boxed{\triangle LMN \sim \triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime}}$[/tex]
### Step-by-Step Solution:
1. Angles under dilation:
- Statement: [tex]$\angle M = \angle M^{\prime \prime}$[/tex].
- Reasoning: Dilation transformations preserve the angles of a shape. This means that the angles of [tex]$\triangle LMN$[/tex] will remain the same in [tex]$\triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime}$[/tex].
- Conclusion: This statement is true.
2. Similarity of triangles:
- Statement: [tex]$\triangle LMN \sim \triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime}$[/tex].
- Reasoning: A dilation transformation scales the sides of a triangle proportionally but does not change the shape of the triangle. This means the triangles are similar by definition.
- Conclusion: This statement is true.
3. Congruence of triangles:
- Statement: [tex]$\triangle LMN = \triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime}$[/tex].
- Reasoning: Congruence requires the triangles to have identical side lengths and identical angles. While the angles remain unchanged, dilation changes the side lengths by a scale factor. Since different scale factors (2 and 0.75) have been applied, the side lengths of the resultant triangle will differ from the original.
- Conclusion: This statement is false.
4. Coordinates of vertices:
Let's find the coordinates of the vertices after applying the given dilations.
- First dilation [tex]\(D_{O,2} (x,y)\)[/tex]:
- Scale factor is 2 (about the origin).
- The coordinates of a point [tex]\((x, y)\)[/tex] will become [tex]\((2x, 2y)\)[/tex].
- Second dilation [tex]\(D_{0,0.75} (x,y)\)[/tex]:
- Scale factor is 0.75 (about the origin).
- The coordinates of a point [tex]\((2x, 2y)\)[/tex] will be scaled down to [tex]\((0.75 \cdot 2x, 0.75 \cdot 2y) = (1.5x, 1.5y)\)[/tex].
The final coordinates of the vertices will be calculated based on their original coordinates.
- [tex]\(\mathbf{L}\)[/tex]:
- Original: [tex]\((-3, 1.5)\)[/tex]
- After [tex]\(D_{O, 2}\)[/tex]: [tex]\((-6, 3)\)[/tex]
- After [tex]\(D_{0, 0.75}\)[/tex]: [tex]\((-6 \cdot 0.75, 3 \cdot 0.75) = (-4.5, 2.25)\)[/tex]
- Given statement says the coordinates are [tex]\((-3, 1.5)\)[/tex].
- Conclusion: This statement is false.
- [tex]\(\mathbf{M}\)[/tex]:
- Original: [tex]\((1.5, -1.5)\)[/tex]
- After [tex]\(D_{O, 2}\)[/tex]: [tex]\((3, -3)\)[/tex]
- After [tex]\(D_{0, 0.75}\)[/tex]: [tex]\((3 \cdot 0.75, -3 \cdot 0.75) = (2.25, -2.25)\)[/tex]
- Given statement says the coordinates are [tex]\((1.5, -1.5)\)[/tex].
- Conclusion: This statement is false.
- [tex]\(\mathbf{N}\)[/tex]:
- Original: [tex]\((3, -1.5)\)[/tex]
- After [tex]\(D_{O, 2}\)[/tex]: [tex]\((6, -3)\)[/tex]
- After [tex]\(D_{0, 0.75}\)[/tex]: [tex]\((6 \cdot 0.75, -3 \cdot 0.75) = (4.5, -2.25)\)[/tex]
- Given statement says the coordinates are [tex]\((3, -1.5)\)[/tex].
- Conclusion: This statement is false.
### Summary:
The true statements regarding the dilations applied to [tex]\(\triangle LMN\)[/tex] are:
- [tex]$\angle M = \angle M^{\prime \prime}$[/tex]
- [tex]$\triangle LMN \sim \triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime}$[/tex]
Hence, the final selection will be:
- [tex]$\boxed{\angle M = \angle M^{\prime \prime}}$[/tex]
- [tex]$\boxed{\triangle LMN \sim \triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime}}$[/tex]
