Answer :
Sure, I'll guide you through the problem step-by-step. Based on the result you have given and treating it as true, I'll work through each statement to verify their validity.
### Step-by-Step Verification:
1. Consider the transformations:
The transformations mentioned in the problem are dilation operations denoted by [tex]\(D_{O, k}(x, y)\)[/tex].
- [tex]\(D_{0,0.75}(x, y)\)[/tex] represents a dilation with the center at the origin and a scale factor of 0.75.
- [tex]\(D_{O, 2}(x, y)\)[/tex] represents a dilation with the center at the origin and a scale factor of 2.
2. Dilation Composition:
To apply this composition of dilations:
- First, the triangle [tex]\(\triangle LMN\)[/tex] is scaled by a factor of 2, which will double its size.
- Then, this resulting triangle is scaled again by a factor of 0.75, which will reduce its size to 75% of the previous size.
3. Calculations:
If the initial coordinates of [tex]\(L\)[/tex], [tex]\(M\)[/tex], and [tex]\(N\)[/tex] are [tex]\((x_1, y_1)\)[/tex], [tex]\((x_2, y_2)\)[/tex], and [tex]\((x_3, y_3)\)[/tex] respectively, then:
- After the first dilation, the new coordinates will be:
[tex]\[ L' = (2x_1, 2y_1), \quad M' = (2x_2, 2y_2), \quad N' = (2x_3, 2y_3) \][/tex]
- After the second dilation, the final coordinates will be:
[tex]\[ L'' = (0.75 \cdot 2x_1, 0.75 \cdot 2y_1) = (1.5x_1, 1.5y_1) \quad M'' = (1.5x_2, 1.5y_2), \quad N'' = (1.5x_3, 1.5y_3) \][/tex]
4. Angles:
Dilation transformations preserve the angles between points since the shape is uniformly scaled. Therefore:
- [tex]\(\angle M \cong \angle M''\)[/tex]
5. Similarity and Congruence:
The original triangle [tex]\(\triangle LMN\)[/tex] and the transformed triangle [tex]\(\triangle L''M''N''\)[/tex] are similar (in fact, they are scaled versions of each other). However, they are not congruent since their sizes differ:
- [tex]\(\triangle LMN \sim \triangle L''M''N''\)[/tex]
- [tex]\(\triangle LMN \not\cong \triangle L''M''N''\)[/tex]
6. Vertex Coordinates:
Considering the true answer from the question's context:
- The coordinates of vertex [tex]\(L''\)[/tex] are [tex]\((-3, 1.5)\)[/tex]
- The coordinates of vertex [tex]\(N''\)[/tex] are [tex]\((3, -1.5)\)[/tex]
- The coordinates of vertex [tex]\(M''\)[/tex] are [tex]\((1.5, -1.5)\)[/tex]
### Verified Statements:
- [tex]\(\angle M \cong \angle M''\)[/tex]
- [tex]\(\triangle LMN \not\cong \triangle L''M''N''\)[/tex]
- The coordinates of vertex [tex]\(L''\)[/tex] are [tex]\((-3, 1.5)\)[/tex]
- The coordinates of vertex [tex]\(N''\)[/tex] are [tex]\((3, -1.5)\)[/tex]
- The coordinates of vertex [tex]\(M''\)[/tex] are [tex]\((1.5, -1.5)\)[/tex]
Hence, the true checked statements are:
- [tex]\(\angle M \cong \angle M''\)[/tex]
- [tex]\(\triangle LMN \not\cong \triangle L''M''N''\)[/tex]
- The coordinates of vertex [tex]\(L''\)[/tex] are [tex]\((-3, 1.5)\)[/tex]
- The coordinates of vertex [tex]\(N''\)[/tex] are [tex]\((3, -1.5)\)[/tex]
- The coordinates of vertex [tex]\(M''\)[/tex] are [tex]\((1.5, -1.5)\)[/tex]
### Step-by-Step Verification:
1. Consider the transformations:
The transformations mentioned in the problem are dilation operations denoted by [tex]\(D_{O, k}(x, y)\)[/tex].
- [tex]\(D_{0,0.75}(x, y)\)[/tex] represents a dilation with the center at the origin and a scale factor of 0.75.
- [tex]\(D_{O, 2}(x, y)\)[/tex] represents a dilation with the center at the origin and a scale factor of 2.
2. Dilation Composition:
To apply this composition of dilations:
- First, the triangle [tex]\(\triangle LMN\)[/tex] is scaled by a factor of 2, which will double its size.
- Then, this resulting triangle is scaled again by a factor of 0.75, which will reduce its size to 75% of the previous size.
3. Calculations:
If the initial coordinates of [tex]\(L\)[/tex], [tex]\(M\)[/tex], and [tex]\(N\)[/tex] are [tex]\((x_1, y_1)\)[/tex], [tex]\((x_2, y_2)\)[/tex], and [tex]\((x_3, y_3)\)[/tex] respectively, then:
- After the first dilation, the new coordinates will be:
[tex]\[ L' = (2x_1, 2y_1), \quad M' = (2x_2, 2y_2), \quad N' = (2x_3, 2y_3) \][/tex]
- After the second dilation, the final coordinates will be:
[tex]\[ L'' = (0.75 \cdot 2x_1, 0.75 \cdot 2y_1) = (1.5x_1, 1.5y_1) \quad M'' = (1.5x_2, 1.5y_2), \quad N'' = (1.5x_3, 1.5y_3) \][/tex]
4. Angles:
Dilation transformations preserve the angles between points since the shape is uniformly scaled. Therefore:
- [tex]\(\angle M \cong \angle M''\)[/tex]
5. Similarity and Congruence:
The original triangle [tex]\(\triangle LMN\)[/tex] and the transformed triangle [tex]\(\triangle L''M''N''\)[/tex] are similar (in fact, they are scaled versions of each other). However, they are not congruent since their sizes differ:
- [tex]\(\triangle LMN \sim \triangle L''M''N''\)[/tex]
- [tex]\(\triangle LMN \not\cong \triangle L''M''N''\)[/tex]
6. Vertex Coordinates:
Considering the true answer from the question's context:
- The coordinates of vertex [tex]\(L''\)[/tex] are [tex]\((-3, 1.5)\)[/tex]
- The coordinates of vertex [tex]\(N''\)[/tex] are [tex]\((3, -1.5)\)[/tex]
- The coordinates of vertex [tex]\(M''\)[/tex] are [tex]\((1.5, -1.5)\)[/tex]
### Verified Statements:
- [tex]\(\angle M \cong \angle M''\)[/tex]
- [tex]\(\triangle LMN \not\cong \triangle L''M''N''\)[/tex]
- The coordinates of vertex [tex]\(L''\)[/tex] are [tex]\((-3, 1.5)\)[/tex]
- The coordinates of vertex [tex]\(N''\)[/tex] are [tex]\((3, -1.5)\)[/tex]
- The coordinates of vertex [tex]\(M''\)[/tex] are [tex]\((1.5, -1.5)\)[/tex]
Hence, the true checked statements are:
- [tex]\(\angle M \cong \angle M''\)[/tex]
- [tex]\(\triangle LMN \not\cong \triangle L''M''N''\)[/tex]
- The coordinates of vertex [tex]\(L''\)[/tex] are [tex]\((-3, 1.5)\)[/tex]
- The coordinates of vertex [tex]\(N''\)[/tex] are [tex]\((3, -1.5)\)[/tex]
- The coordinates of vertex [tex]\(M''\)[/tex] are [tex]\((1.5, -1.5)\)[/tex]
