Answer :
### 1. Part A:
The translation rule [tex]\((x, y) \to (x-4, y+3)\)[/tex] can be described as follows:
- Translation Description: The translation moves each point of the original triangle 4 units to the left (because of [tex]\(x-4\)[/tex]) and 3 units up (because of [tex]\(y+3\)[/tex]).
### 2. Part B:
To find the new vertices of triangle [tex]\(A'B'C'\)[/tex] after applying the translation rule, we follow these steps:
- Given vertices:
- [tex]\(A(-3, 1)\)[/tex]
- [tex]\(B(-3, 4)\)[/tex]
- [tex]\(C(-7, 1)\)[/tex]
- Applying the translation:
- For [tex]\(A(-3, 1)\)[/tex]:
[tex]\[ A': (-3 - 4, 1 + 3) = (-7, 4) \][/tex]
- For [tex]\(B(-3, 4)\)[/tex]:
[tex]\[ B': (-3 - 4, 4 + 3) = (-7, 7) \][/tex]
- For [tex]\(C(-7, 1)\)[/tex]:
[tex]\[ C': (-7 - 4, 1 + 3) = (-11, 4) \][/tex]
- New vertices of [tex]\(A'B'C'\)[/tex]:
- [tex]\(A'(-7, 4)\)[/tex]
- [tex]\(B'(-7, 7)\)[/tex]
- [tex]\(C'(-11, 4)\)[/tex]
### 3. Part C:
To rotate triangle [tex]\(A'B'C'\)[/tex] 90° clockwise about the origin, we apply the following rule for each point [tex]\((x, y)\)[/tex] to get new coordinates [tex]\((y, -x)\)[/tex]:
- Applying the rotation:
- For [tex]\(A'(-7, 4)\)[/tex]:
[tex]\[ A'': (4, -(-7)) = (4, 7) \][/tex]
- For [tex]\(B'(-7, 7)\)[/tex]:
[tex]\[ B'': (7, -(-7)) = (7, 7) \][/tex]
- For [tex]\(C'(-11, 4)\)[/tex]:
[tex]\[ C'': (4, -(-11)) = (4, 11) \][/tex]
- New vertices of [tex]\(A''B''C''\)[/tex]:
- [tex]\(A''(4, 7)\)[/tex]
- [tex]\(B''(7, 7)\)[/tex]
- [tex]\(C''(4, 11)\)[/tex]
- Congruence:
- Triangle [tex]\(ABC\)[/tex] and triangle [tex]\(A''B''C''\)[/tex] are congruent. Translations and rotations are rigid transformations, meaning they preserve the distances and angles of the pre-image. Therefore, the size and shape of triangle [tex]\(ABC\)[/tex] are preserved in triangle [tex]\(A''B''C''\)[/tex], ensuring that they are congruent.
The translation rule [tex]\((x, y) \to (x-4, y+3)\)[/tex] can be described as follows:
- Translation Description: The translation moves each point of the original triangle 4 units to the left (because of [tex]\(x-4\)[/tex]) and 3 units up (because of [tex]\(y+3\)[/tex]).
### 2. Part B:
To find the new vertices of triangle [tex]\(A'B'C'\)[/tex] after applying the translation rule, we follow these steps:
- Given vertices:
- [tex]\(A(-3, 1)\)[/tex]
- [tex]\(B(-3, 4)\)[/tex]
- [tex]\(C(-7, 1)\)[/tex]
- Applying the translation:
- For [tex]\(A(-3, 1)\)[/tex]:
[tex]\[ A': (-3 - 4, 1 + 3) = (-7, 4) \][/tex]
- For [tex]\(B(-3, 4)\)[/tex]:
[tex]\[ B': (-3 - 4, 4 + 3) = (-7, 7) \][/tex]
- For [tex]\(C(-7, 1)\)[/tex]:
[tex]\[ C': (-7 - 4, 1 + 3) = (-11, 4) \][/tex]
- New vertices of [tex]\(A'B'C'\)[/tex]:
- [tex]\(A'(-7, 4)\)[/tex]
- [tex]\(B'(-7, 7)\)[/tex]
- [tex]\(C'(-11, 4)\)[/tex]
### 3. Part C:
To rotate triangle [tex]\(A'B'C'\)[/tex] 90° clockwise about the origin, we apply the following rule for each point [tex]\((x, y)\)[/tex] to get new coordinates [tex]\((y, -x)\)[/tex]:
- Applying the rotation:
- For [tex]\(A'(-7, 4)\)[/tex]:
[tex]\[ A'': (4, -(-7)) = (4, 7) \][/tex]
- For [tex]\(B'(-7, 7)\)[/tex]:
[tex]\[ B'': (7, -(-7)) = (7, 7) \][/tex]
- For [tex]\(C'(-11, 4)\)[/tex]:
[tex]\[ C'': (4, -(-11)) = (4, 11) \][/tex]
- New vertices of [tex]\(A''B''C''\)[/tex]:
- [tex]\(A''(4, 7)\)[/tex]
- [tex]\(B''(7, 7)\)[/tex]
- [tex]\(C''(4, 11)\)[/tex]
- Congruence:
- Triangle [tex]\(ABC\)[/tex] and triangle [tex]\(A''B''C''\)[/tex] are congruent. Translations and rotations are rigid transformations, meaning they preserve the distances and angles of the pre-image. Therefore, the size and shape of triangle [tex]\(ABC\)[/tex] are preserved in triangle [tex]\(A''B''C''\)[/tex], ensuring that they are congruent.
