Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
