From science to arts, IDNLearn.com has the answers to all your questions. Ask your questions and receive comprehensive, trustworthy responses from our dedicated team of experts.
Sagot :
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]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for choosing IDNLearn.com. We’re committed to providing accurate answers, so visit us again soon.
