Answer :
To solve this problem, we need to understand the effects of applying two dilations in succession to a triangle [tex]\( \triangle LMN \)[/tex].
### Step-by-step Analysis:
1. Dilation [tex]\( D_{O, 2}(x, y) \)[/tex]:
- This transformation dilates the triangle [tex]\( \triangle LMN \)[/tex] with a scale factor of 2 about the origin [tex]\( O \)[/tex].
2. Dilation [tex]\( D_{O, 0.75}(x, y) \)[/tex]:
- This further dilates the resulting triangle from the first dilation with a scale factor of 0.75 about the origin [tex]\( O \)[/tex].
The combined effect of these two dilations is equivalent to a single dilation [tex]\( D_{O, 1.5}(x, y) \)[/tex], since [tex]\( 2 \times 0.75 = 1.5 \)[/tex].
### Verifying Statements:
1. [tex]\(\angle M = \angle M^{\prime\prime}\)[/tex]:
- True. Dilations preserve angles, so the angle remains the same.
2. [tex]\(\triangle LMN \sim \triangle L^{\prime\prime}M^{\prime\prime}N^{\prime\prime}\)[/tex]:
- True. The triangles are similar because a dilation changes size but not shape. Therefore, they are geometrically similar triangles.
3. [tex]\(\triangle LMN = \triangle L^{\prime\prime}M^{\prime\prime}N^{\prime\prime}\)[/tex]:
- False. The triangles are not congruent because their sizes differ due to the dilation.
4. Coordinates of [tex]\(L^{\prime \prime}\)[/tex] are [tex]\((-3, 1.5)\)[/tex]:
- True. Given that the final dilation involves a scale of 1.5, if [tex]\(L ( -2 , 1 )\)[/tex] is a vertex of the original triangle, after applying the dilation the coordinates become [tex]\( (-2 \times 1.5, 1 \times 1.5) = (-3, 1.5) \)[/tex].
5. Coordinates of [tex]\(N^{\prime \prime}\)[/tex] are [tex]\((3, -1.5)\)[/tex]:
- True. Given that [tex]\(N (2, -1)\)[/tex] is transformed, after applying the dilation the coordinates become [tex]\( (2 \times 1.5, -1 \times 1.5) = (3, -1.5) \)[/tex].
6. Coordinates of [tex]\(M^{\prime \prime}\)[/tex] are [tex]\((1.5, -1.5)\)[/tex]:
- True. Given [tex]\(M (1, -1)\)[/tex] as a vertex of the original triangle, after applying the dilation the coordinates are [tex]\( (1 \times 1.5, -1 \times 1.5) = (1.5, -1.5)\)[/tex].
### Conclusion:
Based on the above analysis, the statements that must be true are:
- [tex]\(\angle M = \angle M^{\prime \prime}\)[/tex]
- [tex]\(\triangle LMN \sim \triangle L^{\prime\prime} M^{\prime\prime} N^{\prime\prime}\)[/tex]
- The coordinates of vertex [tex]\(L^{\prime\prime}\)[/tex] are [tex]\((-3,1.5)\)[/tex].
- The coordinates of vertex [tex]\(N^{\prime\prime}\)[/tex] are [tex]\((3,-1.5)\)[/tex].
- The coordinates of vertex [tex]\(M^{\prime\prime}\)[/tex] are [tex]\((1.5,-1.5)\)[/tex].
### Step-by-step Analysis:
1. Dilation [tex]\( D_{O, 2}(x, y) \)[/tex]:
- This transformation dilates the triangle [tex]\( \triangle LMN \)[/tex] with a scale factor of 2 about the origin [tex]\( O \)[/tex].
2. Dilation [tex]\( D_{O, 0.75}(x, y) \)[/tex]:
- This further dilates the resulting triangle from the first dilation with a scale factor of 0.75 about the origin [tex]\( O \)[/tex].
The combined effect of these two dilations is equivalent to a single dilation [tex]\( D_{O, 1.5}(x, y) \)[/tex], since [tex]\( 2 \times 0.75 = 1.5 \)[/tex].
### Verifying Statements:
1. [tex]\(\angle M = \angle M^{\prime\prime}\)[/tex]:
- True. Dilations preserve angles, so the angle remains the same.
2. [tex]\(\triangle LMN \sim \triangle L^{\prime\prime}M^{\prime\prime}N^{\prime\prime}\)[/tex]:
- True. The triangles are similar because a dilation changes size but not shape. Therefore, they are geometrically similar triangles.
3. [tex]\(\triangle LMN = \triangle L^{\prime\prime}M^{\prime\prime}N^{\prime\prime}\)[/tex]:
- False. The triangles are not congruent because their sizes differ due to the dilation.
4. Coordinates of [tex]\(L^{\prime \prime}\)[/tex] are [tex]\((-3, 1.5)\)[/tex]:
- True. Given that the final dilation involves a scale of 1.5, if [tex]\(L ( -2 , 1 )\)[/tex] is a vertex of the original triangle, after applying the dilation the coordinates become [tex]\( (-2 \times 1.5, 1 \times 1.5) = (-3, 1.5) \)[/tex].
5. Coordinates of [tex]\(N^{\prime \prime}\)[/tex] are [tex]\((3, -1.5)\)[/tex]:
- True. Given that [tex]\(N (2, -1)\)[/tex] is transformed, after applying the dilation the coordinates become [tex]\( (2 \times 1.5, -1 \times 1.5) = (3, -1.5) \)[/tex].
6. Coordinates of [tex]\(M^{\prime \prime}\)[/tex] are [tex]\((1.5, -1.5)\)[/tex]:
- True. Given [tex]\(M (1, -1)\)[/tex] as a vertex of the original triangle, after applying the dilation the coordinates are [tex]\( (1 \times 1.5, -1 \times 1.5) = (1.5, -1.5)\)[/tex].
### Conclusion:
Based on the above analysis, the statements that must be true are:
- [tex]\(\angle M = \angle M^{\prime \prime}\)[/tex]
- [tex]\(\triangle LMN \sim \triangle L^{\prime\prime} M^{\prime\prime} N^{\prime\prime}\)[/tex]
- The coordinates of vertex [tex]\(L^{\prime\prime}\)[/tex] are [tex]\((-3,1.5)\)[/tex].
- The coordinates of vertex [tex]\(N^{\prime\prime}\)[/tex] are [tex]\((3,-1.5)\)[/tex].
- The coordinates of vertex [tex]\(M^{\prime\prime}\)[/tex] are [tex]\((1.5,-1.5)\)[/tex].
