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Triangle ABC has vertices [tex]\(A(-3, 1)\)[/tex], [tex]\(B(-3, 4)\)[/tex], and [tex]\(C(-7, 1)\)[/tex].

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

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



Answer :

GinnyAnswer
Certainly! Let's break down and solve the problem step by step.

### Part A: Translation Description

The translation given is according to the rule (x, y) → (x - 3, y + 4). This translation can be described in words as follows:

- Move each point 3 units to the left.
- Move each point 4 units up.

### Part B: Calculating New Coordinates

Next, we need to determine the new coordinates of the vertices A', B', and C' after applying the translation.

The original coordinates of the vertices are:
- A: (-3, 1)
- B: (-3, 4)
- C: (-7, 1)

#### Applying the Translation to Each Vertex:

1. Vertex A(-3, 1):
- New x-coordinate: -3 - 3 = -6
- New y-coordinate: 1 + 4 = 5
- Therefore, A' is at (-6, 5).

2. Vertex B(-3, 4):
- New x-coordinate: -3 - 3 = -6
- New y-coordinate: 4 + 4 = 8
- Therefore, B' is at (-6, 8).

3. Vertex C(-7, 1):
- New x-coordinate: -7 - 3 = -10
- New y-coordinate: 1 + 4 = 5
- Therefore, C' is at (-10, 5).

### Final Coordinates for AA'B'C':
- A' is at (-6, 5)
- B' is at (-6, 8)
- C' is at (-10, 5)

So, the vertices of the translated triangle A'B'C' are located at (-6, 5), (-6, 8), and (-10, 5).