Answer :
Certainly! Let's solve each part of the problem step-by-step.
### Part A: Translation Description
When a point [tex]\((x, y)\)[/tex] is translated according to the rule [tex]\((x-4, y+3)\)[/tex], it means:
- The [tex]\(x\)[/tex]-coordinate of each point is decreased by 4 units.
- The [tex]\(y\)[/tex]-coordinate of each point is increased by 3 units.
In words, the translation can be described as:
“Each point is shifted 4 units to the left and 3 units up.”
### Part B: Coordinates of Translated Triangle [tex]\(A'B'C'\)[/tex]
To find the coordinates of [tex]\(\triangle A'B'C'\)[/tex], we apply the translation [tex]\((x-4, y+3)\)[/tex] to each vertex of [tex]\(\triangle ABC\)[/tex].
1. Vertex [tex]\(A\)[/tex]:
- Original: [tex]\(A(-3, 1)\)[/tex]
- Translated: [tex]\(A' = (-3 - 4, 1 + 3) = (-7, 4)\)[/tex]
2. Vertex [tex]\(B\)[/tex]:
- Original: [tex]\(B(-3, 4)\)[/tex]
- Translated: [tex]\(B' = (-3 - 4, 4 + 3) = (-7, 7)\)[/tex]
3. Vertex [tex]\(C\)[/tex]:
- Original: [tex]\(C(-7, 1)\)[/tex]
- Translated: [tex]\(C' = (-7 - 4, 1 + 3) = (-11, 4)\)[/tex]
So, the coordinates of the vertices of [tex]\(\triangle A'B'C'\)[/tex] are:
- [tex]\(A'(-7, 4)\)[/tex]
- [tex]\(B'(-7, 7)\)[/tex]
- [tex]\(C'(-11, 4)\)[/tex]
### Part C: Rotation of [tex]\(\triangle A'B'C'\)[/tex] to Form [tex]\(\triangle A''B''C''\)[/tex]
When [tex]\(\triangle A'B'C'\)[/tex] is rotated [tex]\(90^\circ\)[/tex] clockwise about the origin, each point [tex]\((x, y)\)[/tex] is transformed according to the rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex].
1. Vertex [tex]\(A'\)[/tex]:
- Original: [tex]\(A'(-7, 4)\)[/tex]
- Rotated: [tex]\(A'' = (4, 7)\)[/tex]
2. Vertex [tex]\(B'\)[/tex]:
- Original: [tex]\(B'(-7, 7)\)[/tex]
- Rotated: [tex]\(B'' = (7, 7)\)[/tex]
3. Vertex [tex]\(C'\)[/tex]:
- Original: [tex]\(C'(-11, 4)\)[/tex]
- Rotated: [tex]\(C'' = (4, 11)\)[/tex]
So, the coordinates of the vertices of [tex]\(\triangle A''B''C''\)[/tex] are:
- [tex]\(A''(4, 7)\)[/tex]
- [tex]\(B''(7, 7)\)[/tex]
- [tex]\(C''(4, 11)\)[/tex]
### Congruency Check between [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle A''B''C''\)[/tex]
To determine if [tex]\(\triangle ABC\)[/tex] is congruent to [tex]\(\triangle A''B''C''\)[/tex], we should check if the side lengths of each triangle are preserved through the transformations.
1. Length of [tex]\(AB\)[/tex]:
[tex]\[ AB = \sqrt{(-3 - (-3))^2 + (4 - 1)^2} = \sqrt{0 + 3^2} = 3 \][/tex]
2. Length of [tex]\(BC\)[/tex]:
[tex]\[ BC = \sqrt{(-3 - (-7))^2 + (4 - 1)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5 \][/tex]
3. Length of [tex]\(AC\)[/tex]:
[tex]\[ AC = \sqrt{(-3 - (-7))^2 + (1 - 1)^2} = \sqrt{4^2 + 0} = 4 \][/tex]
4. Length of [tex]\(A''B''\)[/tex]:
[tex]\[ A''B'' = \sqrt{(4 - 7)^2 + (7 - 7)^2} = \sqrt{(-3)^2 + 0} = 3 \][/tex]
5. Length of [tex]\(B''C''\)[/tex]:
[tex]\[ B''C'' = \sqrt{(7 - 4)^2 + (7 - 11)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = 5 \][/tex]
6. Length of [tex]\(A''C''\)[/tex]:
[tex]\[ A''C'' = \sqrt{(4 - 4)^2 + (7 - 11)^2} = \sqrt{0 + (-4)^2} = 4 \][/tex]
Since the lengths of the sides are preserved:
- [tex]\(AB = A''B'' = 3\)[/tex]
- [tex]\(BC = B''C'' = 5\)[/tex]
- [tex]\(AC = A''C'' = 4\)[/tex]
[tex]\(\triangle ABC\)[/tex] is congruent to [tex]\(\triangle A''B''C''\)[/tex].
### Conclusion
Yes, [tex]\(\triangle ABC\)[/tex] is congruent to [tex]\(\triangle A''B''C''\)[/tex] because the lengths of the sides of the triangles are equal, preserving the triangles' shapes and sizes through the translation and rotation processes.
### Part A: Translation Description
When a point [tex]\((x, y)\)[/tex] is translated according to the rule [tex]\((x-4, y+3)\)[/tex], it means:
- The [tex]\(x\)[/tex]-coordinate of each point is decreased by 4 units.
- The [tex]\(y\)[/tex]-coordinate of each point is increased by 3 units.
In words, the translation can be described as:
“Each point is shifted 4 units to the left and 3 units up.”
### Part B: Coordinates of Translated Triangle [tex]\(A'B'C'\)[/tex]
To find the coordinates of [tex]\(\triangle A'B'C'\)[/tex], we apply the translation [tex]\((x-4, y+3)\)[/tex] to each vertex of [tex]\(\triangle ABC\)[/tex].
1. Vertex [tex]\(A\)[/tex]:
- Original: [tex]\(A(-3, 1)\)[/tex]
- Translated: [tex]\(A' = (-3 - 4, 1 + 3) = (-7, 4)\)[/tex]
2. Vertex [tex]\(B\)[/tex]:
- Original: [tex]\(B(-3, 4)\)[/tex]
- Translated: [tex]\(B' = (-3 - 4, 4 + 3) = (-7, 7)\)[/tex]
3. Vertex [tex]\(C\)[/tex]:
- Original: [tex]\(C(-7, 1)\)[/tex]
- Translated: [tex]\(C' = (-7 - 4, 1 + 3) = (-11, 4)\)[/tex]
So, the coordinates of the vertices of [tex]\(\triangle A'B'C'\)[/tex] are:
- [tex]\(A'(-7, 4)\)[/tex]
- [tex]\(B'(-7, 7)\)[/tex]
- [tex]\(C'(-11, 4)\)[/tex]
### Part C: Rotation of [tex]\(\triangle A'B'C'\)[/tex] to Form [tex]\(\triangle A''B''C''\)[/tex]
When [tex]\(\triangle A'B'C'\)[/tex] is rotated [tex]\(90^\circ\)[/tex] clockwise about the origin, each point [tex]\((x, y)\)[/tex] is transformed according to the rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex].
1. Vertex [tex]\(A'\)[/tex]:
- Original: [tex]\(A'(-7, 4)\)[/tex]
- Rotated: [tex]\(A'' = (4, 7)\)[/tex]
2. Vertex [tex]\(B'\)[/tex]:
- Original: [tex]\(B'(-7, 7)\)[/tex]
- Rotated: [tex]\(B'' = (7, 7)\)[/tex]
3. Vertex [tex]\(C'\)[/tex]:
- Original: [tex]\(C'(-11, 4)\)[/tex]
- Rotated: [tex]\(C'' = (4, 11)\)[/tex]
So, the coordinates of the vertices of [tex]\(\triangle A''B''C''\)[/tex] are:
- [tex]\(A''(4, 7)\)[/tex]
- [tex]\(B''(7, 7)\)[/tex]
- [tex]\(C''(4, 11)\)[/tex]
### Congruency Check between [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle A''B''C''\)[/tex]
To determine if [tex]\(\triangle ABC\)[/tex] is congruent to [tex]\(\triangle A''B''C''\)[/tex], we should check if the side lengths of each triangle are preserved through the transformations.
1. Length of [tex]\(AB\)[/tex]:
[tex]\[ AB = \sqrt{(-3 - (-3))^2 + (4 - 1)^2} = \sqrt{0 + 3^2} = 3 \][/tex]
2. Length of [tex]\(BC\)[/tex]:
[tex]\[ BC = \sqrt{(-3 - (-7))^2 + (4 - 1)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5 \][/tex]
3. Length of [tex]\(AC\)[/tex]:
[tex]\[ AC = \sqrt{(-3 - (-7))^2 + (1 - 1)^2} = \sqrt{4^2 + 0} = 4 \][/tex]
4. Length of [tex]\(A''B''\)[/tex]:
[tex]\[ A''B'' = \sqrt{(4 - 7)^2 + (7 - 7)^2} = \sqrt{(-3)^2 + 0} = 3 \][/tex]
5. Length of [tex]\(B''C''\)[/tex]:
[tex]\[ B''C'' = \sqrt{(7 - 4)^2 + (7 - 11)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = 5 \][/tex]
6. Length of [tex]\(A''C''\)[/tex]:
[tex]\[ A''C'' = \sqrt{(4 - 4)^2 + (7 - 11)^2} = \sqrt{0 + (-4)^2} = 4 \][/tex]
Since the lengths of the sides are preserved:
- [tex]\(AB = A''B'' = 3\)[/tex]
- [tex]\(BC = B''C'' = 5\)[/tex]
- [tex]\(AC = A''C'' = 4\)[/tex]
[tex]\(\triangle ABC\)[/tex] is congruent to [tex]\(\triangle A''B''C''\)[/tex].
### Conclusion
Yes, [tex]\(\triangle ABC\)[/tex] is congruent to [tex]\(\triangle A''B''C''\)[/tex] because the lengths of the sides of the triangles are equal, preserving the triangles' shapes and sizes through the translation and rotation processes.
