Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
Sure, let's solve the problem step by step.
### Part A: Translation Description
Given the translation rule [tex]\((x, y) \rightarrow (x - 4, y + 3)\)[/tex], we need to describe this in words.
Answer:
The translation can be described as "moving each point 4 units to the left and 3 units up."
### Part B: Determining the Vertices of [tex]\(\triangle A'B'C'\)[/tex]
To find the new coordinates of the vertices after translation, we apply the translation rule to each vertex of triangle [tex]\(ABC\)[/tex].
1. For vertex [tex]\(A(-3, 1)\)[/tex]:
[tex]\[ A' = (-3 - 4, 1 + 3) = (-7, 4) \][/tex]
2. For vertex [tex]\(B(-3, 4)\)[/tex]:
[tex]\[ B' = (-3 - 4, 4 + 3) = (-7, 7) \][/tex]
3. For vertex [tex]\(C(-7, 1)\)[/tex]:
[tex]\[ C' = (-7 - 4, 1 + 3) = (-11, 4) \][/tex]
Answer:
The vertices of [tex]\(\triangle A'B'C'\)[/tex] are located at:
- [tex]\(A'(-7, 4)\)[/tex]
- [tex]\(B'(-7, 7)\)[/tex]
- [tex]\(C'(-11, 4)\)[/tex]
### Part C: Rotation and Congruency
To rotate [tex]\(\triangle A'B'C'\)[/tex] 90 degrees clockwise about the origin, we use the rotation rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex].
1. For vertex [tex]\(A'(-7, 4)\)[/tex]:
[tex]\[ A'' = (4, -(-7)) = (4, 7) \][/tex]
2. For vertex [tex]\(B'(-7, 7)\)[/tex]:
[tex]\[ B'' = (7, -(-7)) = (7, 7) \][/tex]
3. For vertex [tex]\(C'(-11, 4)\)[/tex]:
[tex]\[ C'' = (4, -(-11)) = (4, 11) \][/tex]
Are [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle A'' B'' C''\)[/tex] congruent?
To check for congruency, we need to compare the side lengths of both triangles. The side lengths of [tex]\(\triangle ABC\)[/tex] are:
1. Side [tex]\(AB\)[/tex]:
[tex]\[ AB = \sqrt{((-3) - (-3))^2 + (4 - 1)^2} = \sqrt{0 + 3^2} = 3 \][/tex]
2. Side [tex]\(BC\)[/tex]:
[tex]\[ BC = \sqrt{((-7) - (-3))^2 + (1 - 4)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = 5 \][/tex]
3. Side [tex]\(CA\)[/tex]:
[tex]\[ CA = \sqrt{((-3) - (-7))^2 + (1 - 1)^2} = \sqrt{4^2 + 0} = 4 \][/tex]
The side lengths of [tex]\(\triangle A''B''C''\)[/tex] are the same as the side lengths of [tex]\(\triangle A'B'C'\)[/tex], since translation and rotation are rigid transformations (they don’t change side lengths).
Answer:
Since the side lengths of [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle A''B''C''\)[/tex] are the same ([tex]\(AB = A''B'' = 3\)[/tex], [tex]\(BC = B''C'' = 5\)[/tex], [tex]\(CA = C''A'' = 4\)[/tex]), we can conclude that:
[tex]\[ \triangle ABC \text{ is congruent to } \triangle A''B''C'' \][/tex]
Hence, [tex]\(\triangle A B C\)[/tex] is congruent to [tex]\(\triangle A'' B'' C''\)[/tex].
### Part A: Translation Description
Given the translation rule [tex]\((x, y) \rightarrow (x - 4, y + 3)\)[/tex], we need to describe this in words.
Answer:
The translation can be described as "moving each point 4 units to the left and 3 units up."
### Part B: Determining the Vertices of [tex]\(\triangle A'B'C'\)[/tex]
To find the new coordinates of the vertices after translation, we apply the translation rule to each vertex of triangle [tex]\(ABC\)[/tex].
1. For vertex [tex]\(A(-3, 1)\)[/tex]:
[tex]\[ A' = (-3 - 4, 1 + 3) = (-7, 4) \][/tex]
2. For vertex [tex]\(B(-3, 4)\)[/tex]:
[tex]\[ B' = (-3 - 4, 4 + 3) = (-7, 7) \][/tex]
3. For vertex [tex]\(C(-7, 1)\)[/tex]:
[tex]\[ C' = (-7 - 4, 1 + 3) = (-11, 4) \][/tex]
Answer:
The vertices of [tex]\(\triangle A'B'C'\)[/tex] are located at:
- [tex]\(A'(-7, 4)\)[/tex]
- [tex]\(B'(-7, 7)\)[/tex]
- [tex]\(C'(-11, 4)\)[/tex]
### Part C: Rotation and Congruency
To rotate [tex]\(\triangle A'B'C'\)[/tex] 90 degrees clockwise about the origin, we use the rotation rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex].
1. For vertex [tex]\(A'(-7, 4)\)[/tex]:
[tex]\[ A'' = (4, -(-7)) = (4, 7) \][/tex]
2. For vertex [tex]\(B'(-7, 7)\)[/tex]:
[tex]\[ B'' = (7, -(-7)) = (7, 7) \][/tex]
3. For vertex [tex]\(C'(-11, 4)\)[/tex]:
[tex]\[ C'' = (4, -(-11)) = (4, 11) \][/tex]
Are [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle A'' B'' C''\)[/tex] congruent?
To check for congruency, we need to compare the side lengths of both triangles. The side lengths of [tex]\(\triangle ABC\)[/tex] are:
1. Side [tex]\(AB\)[/tex]:
[tex]\[ AB = \sqrt{((-3) - (-3))^2 + (4 - 1)^2} = \sqrt{0 + 3^2} = 3 \][/tex]
2. Side [tex]\(BC\)[/tex]:
[tex]\[ BC = \sqrt{((-7) - (-3))^2 + (1 - 4)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = 5 \][/tex]
3. Side [tex]\(CA\)[/tex]:
[tex]\[ CA = \sqrt{((-3) - (-7))^2 + (1 - 1)^2} = \sqrt{4^2 + 0} = 4 \][/tex]
The side lengths of [tex]\(\triangle A''B''C''\)[/tex] are the same as the side lengths of [tex]\(\triangle A'B'C'\)[/tex], since translation and rotation are rigid transformations (they don’t change side lengths).
Answer:
Since the side lengths of [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle A''B''C''\)[/tex] are the same ([tex]\(AB = A''B'' = 3\)[/tex], [tex]\(BC = B''C'' = 5\)[/tex], [tex]\(CA = C''A'' = 4\)[/tex]), we can conclude that:
[tex]\[ \triangle ABC \text{ is congruent to } \triangle A''B''C'' \][/tex]
Hence, [tex]\(\triangle A B C\)[/tex] is congruent to [tex]\(\triangle A'' B'' C''\)[/tex].
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
