Question 1 (Essay Worth 10 points)

Triangle ABC has vertices A(-3, 1), B(-3, 4), and C(-7, 1).

1. Part A: If [tex]\(\triangle ABC\)[/tex] is translated according to the rule [tex]\((x, y) \rightarrow (x-4, y+3)\)[/tex] to form [tex]\(\triangle A'B'C'\)[/tex], how is the translation described in words? (3 points)

2. Part B: Where are the vertices of [tex]\(\triangle A'B'C'\)[/tex] located? Show your work or explain your steps. (4 points)

3. Part C: Triangle [tex]\(\triangle A'B'C'\)[/tex] is rotated 90° clockwise about the origin to form [tex]\(\triangle A''B''C''\)[/tex]. Is [tex]\(\triangle ABC\)[/tex] congruent to [tex]\(\triangle A''B''C''\)[/tex]? Give details to support your answer. (3 points)

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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.