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Sagot :
To solve the problem, we need to apply the composition of two dilations to each vertex of [tex]\(\triangle LMN\)[/tex] and then verify several statements regarding the resulting triangle [tex]\(\triangle L^{-} M^{-} N^{-}\)[/tex].
### Step-by-Step Solution:
1. Understanding the Dilations [tex]\(D_{0,0.75}\)[/tex] and [tex]\(D_{0,2}\)[/tex]:
- The dilation [tex]\(D_{0,0.75}\)[/tex] scales all coordinates by a factor of 0.75.
- The dilation [tex]\(D_{0,2}\)[/tex] scales all coordinates by a factor of 2.
2. Composition of Dilations:
- The overall effect of applying [tex]\(D_{0,0.75}\)[/tex] followed by [tex]\(D_{0,2}\)[/tex] is a single dilation where the factor is the product of the two individual factors:
[tex]\[ \text{Final Factor} = 0.75 \times 2 = 1.5 \][/tex]
3. Applying the Composite Dilation to Each Vertex:
- Original coordinates of [tex]\(L\)[/tex]: [tex]\((-3, 1.5)\)[/tex]
[tex]\[ L^{-} = (-3 \times 1.5, 1.5 \times 1.5) = (-4.5, 2.25) \][/tex]
- Original coordinates of [tex]\(N\)[/tex]: [tex]\((3, -1.5)\)[/tex]
[tex]\[ N^{-} = (3 \times 1.5, -1.5 \times 1.5) = (4.5, -2.25) \][/tex]
- Original coordinates of [tex]\(M\)[/tex]: [tex]\((1.5, -1.5)\)[/tex]
[tex]\[ M^{-} = (1.5 \times 1.5, -1.5 \times 1.5) = (2.25, -2.25) \][/tex]
4. Verification of Statements:
- [tex]\(\angle M \cong \angle M^{-}\)[/tex]:
- True, because dilations preserve the angles of a triangle. Therefore, all corresponding angles in the two triangles will be congruent.
- [tex]\(\triangle LMN \simeq \triangle L^{-} M^{-} N^{-}\)[/tex]:
- False, because dilation changes the side lengths, even though it preserves the shape. Therefore, the triangles are similar but not congruent.
- The coordinates of vertex [tex]\(L^{-}\)[/tex] are [tex]\((-3, 1.5)\)[/tex]:
- False, the correct coordinates after dilation are [tex]\((-4.5, 2.25)\)[/tex].
- The coordinates of vertex [tex]\(N^{-}\)[/tex] are [tex]\((3, -1.5)\)[/tex]:
- False, the correct coordinates after dilation are [tex]\((4.5, -2.25)\)[/tex].
- The coordinates of vertex [tex]\(M^{-}\)[/tex] are [tex]\((1.5, -1.5)\)[/tex]:
- False, the correct coordinates after dilation are [tex]\((2.25, -2.25)\)[/tex].
### Conclusion:
Based on the calculations, the correct statements are:
1. [tex]\(\angle M \cong \angle M^{-}\)[/tex] (True)
2. [tex]\(\triangle LMN \simeq \triangle L^{-} M^{-} N^{-}\)[/tex] (False)
3. The coordinates of vertex [tex]\(L^{-}\)[/tex] are [tex]\((-3, 1.5)\)[/tex] (False)
4. The coordinates of vertex [tex]\(N^{-}\)[/tex] are [tex]\((3, -1.5)\)[/tex] (False)
5. The coordinates of vertex [tex]\(M^{-}\)[/tex] are [tex]\((1.5, -1.5)\)[/tex] (False)
Thus, the final answer is:
[tex]\[ (\text{True}, \text{False}, \text{False}, \text{False}, \text{False}) \][/tex]
### Step-by-Step Solution:
1. Understanding the Dilations [tex]\(D_{0,0.75}\)[/tex] and [tex]\(D_{0,2}\)[/tex]:
- The dilation [tex]\(D_{0,0.75}\)[/tex] scales all coordinates by a factor of 0.75.
- The dilation [tex]\(D_{0,2}\)[/tex] scales all coordinates by a factor of 2.
2. Composition of Dilations:
- The overall effect of applying [tex]\(D_{0,0.75}\)[/tex] followed by [tex]\(D_{0,2}\)[/tex] is a single dilation where the factor is the product of the two individual factors:
[tex]\[ \text{Final Factor} = 0.75 \times 2 = 1.5 \][/tex]
3. Applying the Composite Dilation to Each Vertex:
- Original coordinates of [tex]\(L\)[/tex]: [tex]\((-3, 1.5)\)[/tex]
[tex]\[ L^{-} = (-3 \times 1.5, 1.5 \times 1.5) = (-4.5, 2.25) \][/tex]
- Original coordinates of [tex]\(N\)[/tex]: [tex]\((3, -1.5)\)[/tex]
[tex]\[ N^{-} = (3 \times 1.5, -1.5 \times 1.5) = (4.5, -2.25) \][/tex]
- Original coordinates of [tex]\(M\)[/tex]: [tex]\((1.5, -1.5)\)[/tex]
[tex]\[ M^{-} = (1.5 \times 1.5, -1.5 \times 1.5) = (2.25, -2.25) \][/tex]
4. Verification of Statements:
- [tex]\(\angle M \cong \angle M^{-}\)[/tex]:
- True, because dilations preserve the angles of a triangle. Therefore, all corresponding angles in the two triangles will be congruent.
- [tex]\(\triangle LMN \simeq \triangle L^{-} M^{-} N^{-}\)[/tex]:
- False, because dilation changes the side lengths, even though it preserves the shape. Therefore, the triangles are similar but not congruent.
- The coordinates of vertex [tex]\(L^{-}\)[/tex] are [tex]\((-3, 1.5)\)[/tex]:
- False, the correct coordinates after dilation are [tex]\((-4.5, 2.25)\)[/tex].
- The coordinates of vertex [tex]\(N^{-}\)[/tex] are [tex]\((3, -1.5)\)[/tex]:
- False, the correct coordinates after dilation are [tex]\((4.5, -2.25)\)[/tex].
- The coordinates of vertex [tex]\(M^{-}\)[/tex] are [tex]\((1.5, -1.5)\)[/tex]:
- False, the correct coordinates after dilation are [tex]\((2.25, -2.25)\)[/tex].
### Conclusion:
Based on the calculations, the correct statements are:
1. [tex]\(\angle M \cong \angle M^{-}\)[/tex] (True)
2. [tex]\(\triangle LMN \simeq \triangle L^{-} M^{-} N^{-}\)[/tex] (False)
3. The coordinates of vertex [tex]\(L^{-}\)[/tex] are [tex]\((-3, 1.5)\)[/tex] (False)
4. The coordinates of vertex [tex]\(N^{-}\)[/tex] are [tex]\((3, -1.5)\)[/tex] (False)
5. The coordinates of vertex [tex]\(M^{-}\)[/tex] are [tex]\((1.5, -1.5)\)[/tex] (False)
Thus, the final answer is:
[tex]\[ (\text{True}, \text{False}, \text{False}, \text{False}, \text{False}) \][/tex]
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