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When we apply the composition of transformations to a triangle [tex]\( \triangle LMN \)[/tex] and obtain [tex]\( \triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime} \)[/tex], we should analyze each statement to determine its validity.
1. Angle Preservation:
- [tex]\( \angle M = \angle M^{\prime\prime} \)[/tex]
- True. Dilations preserve angles. Since dilation only changes the size of the triangle without altering the angles, [tex]\( \angle M \)[/tex] remains equal to [tex]\( \angle M^{\prime\prime} \)[/tex].
2. Triangles are Similar:
- [tex]\( \triangle LMN \sim \triangle L^{\prime\prime} M^{\prime\prime} N^{\prime\prime} \)[/tex]
- True. Dilations result in similar triangles because they preserve the shape of the triangle while changing its size. As a result, [tex]\( \triangle LMN \)[/tex] is similar to [tex]\( \triangle L^{\prime\prime} M^{\prime\prime} N^{\prime\prime} \)[/tex].
3. Triangles are Congruent:
- [tex]\( \triangle LMN = \triangle L^{\prime\prime} M^{\prime\prime} N^{\prime\prime} \)[/tex]
- False. Congruence means the triangles have the same size and shape. Since the dilation factors involved (0.75 and 2) combine to a factor of 1.5, the size of [tex]\( \triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime} \)[/tex] is 1.5 times the size of [tex]\( \triangle LMN \)[/tex]. Therefore, they are not congruent.
4. Coordinates of Vertex [tex]\( L^{\prime\prime} \)[/tex]:
- The coordinates of vertex [tex]\( L^{\prime \prime} \)[/tex] are [tex]\( (-3, 1.5) \)[/tex]
- True. The coordinates given for [tex]\( L^{\prime\prime} \)[/tex] as [tex]\( (-3, 1.5) \)[/tex] match the result after applying the composite dilation.
5. Coordinates of Vertex [tex]\( N^{\prime\prime} \)[/tex]:
- The coordinates of vertex [tex]\( N^{\prime \prime} \)[/tex] are [tex]\( (3, -1.5) \)[/tex]
- True. The coordinates given for [tex]\( N^{\prime\prime} \)[/tex] as [tex]\( (3, -1.5) \)[/tex] match the result after applying the composite dilation.
6. Coordinates of Vertex [tex]\( M^{\prime\prime} \)[/tex]:
- The coordinates of vertex [tex]\( M^{\prime \prime} \)[/tex] are [tex]\( (1.5, -1.5) \)[/tex]
- True. The coordinates given for [tex]\( M^{\prime\prime} \)[/tex] as [tex]\( (1.5, -1.5) \)[/tex] match the result after applying the composite dilation.
Therefore, 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]
The statement that must be false is:
- [tex]\( \triangle LMN = \triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime} \)[/tex]
1. Angle Preservation:
- [tex]\( \angle M = \angle M^{\prime\prime} \)[/tex]
- True. Dilations preserve angles. Since dilation only changes the size of the triangle without altering the angles, [tex]\( \angle M \)[/tex] remains equal to [tex]\( \angle M^{\prime\prime} \)[/tex].
2. Triangles are Similar:
- [tex]\( \triangle LMN \sim \triangle L^{\prime\prime} M^{\prime\prime} N^{\prime\prime} \)[/tex]
- True. Dilations result in similar triangles because they preserve the shape of the triangle while changing its size. As a result, [tex]\( \triangle LMN \)[/tex] is similar to [tex]\( \triangle L^{\prime\prime} M^{\prime\prime} N^{\prime\prime} \)[/tex].
3. Triangles are Congruent:
- [tex]\( \triangle LMN = \triangle L^{\prime\prime} M^{\prime\prime} N^{\prime\prime} \)[/tex]
- False. Congruence means the triangles have the same size and shape. Since the dilation factors involved (0.75 and 2) combine to a factor of 1.5, the size of [tex]\( \triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime} \)[/tex] is 1.5 times the size of [tex]\( \triangle LMN \)[/tex]. Therefore, they are not congruent.
4. Coordinates of Vertex [tex]\( L^{\prime\prime} \)[/tex]:
- The coordinates of vertex [tex]\( L^{\prime \prime} \)[/tex] are [tex]\( (-3, 1.5) \)[/tex]
- True. The coordinates given for [tex]\( L^{\prime\prime} \)[/tex] as [tex]\( (-3, 1.5) \)[/tex] match the result after applying the composite dilation.
5. Coordinates of Vertex [tex]\( N^{\prime\prime} \)[/tex]:
- The coordinates of vertex [tex]\( N^{\prime \prime} \)[/tex] are [tex]\( (3, -1.5) \)[/tex]
- True. The coordinates given for [tex]\( N^{\prime\prime} \)[/tex] as [tex]\( (3, -1.5) \)[/tex] match the result after applying the composite dilation.
6. Coordinates of Vertex [tex]\( M^{\prime\prime} \)[/tex]:
- The coordinates of vertex [tex]\( M^{\prime \prime} \)[/tex] are [tex]\( (1.5, -1.5) \)[/tex]
- True. The coordinates given for [tex]\( M^{\prime\prime} \)[/tex] as [tex]\( (1.5, -1.5) \)[/tex] match the result after applying the composite dilation.
Therefore, 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]
The statement that must be false is:
- [tex]\( \triangle LMN = \triangle L^{\prime \prime} M^{\prime \prime} N^{\prime \prime} \)[/tex]
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