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Question 1 (Essay Worth 10 points)

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

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

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

3. Part C: Triangle A'B'C' is rotated 180° counterclockwise about the origin to form triangle A"B"C". Is triangle ABC congruent to triangle A'B'C'? Give details to support your answer. (3 points)



Answer :

GinnyAnswer
### Part A: Translating Triangle ABC (3 points)

The translation rule given is [tex]\((x, y) \rightarrow (x - 3, y + 4)\)[/tex]. In words, this rule can be described as follows: Each vertex of the triangle is shifted 3 units to the left and 4 units upward.

### Part B: Finding the Vertices of Triangle A'A'B'C' (4 points)

To find the new coordinates of each vertex after translation, we will apply the translation rule [tex]\((x - 3, y + 4)\)[/tex] to each original vertex.

- Vertex A: Original coordinates are [tex]\((-3, 1)\)[/tex].
[tex]\[ A' = (-3 - 3, 1 + 4) = (-6, 5) \][/tex]

- Vertex B: Original coordinates are [tex]\((-3, 4)\)[/tex].
[tex]\[ B' = (-3 - 3, 4 + 4) = (-6, 8) \][/tex]

- Vertex C: Original coordinates are [tex]\((-7, 1)\)[/tex].
[tex]\[ C' = (-7 - 3, 1 + 4) = (-10, 5) \][/tex]

Thus, the vertices of [tex]\(\Delta A'B'C'\)[/tex] are:
[tex]\[ A'(-6, 5), B'(-6, 8), C'(-10, 5) \][/tex]

### Part C: Rotating Triangle A'B'C' 180 Degrees Counterclockwise (3 points)

To perform a rotation of 180 degrees counterclockwise about the origin, we use the rotation rule [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]. Applying this rule to each of the vertices of [tex]\(\Delta A'B'C'\)[/tex]:

- Vertex A': Coordinates are [tex]\((-6, 5)\)[/tex].
[tex]\[ A'' = (-(-6), -(5)) = (6, -5) \][/tex]

- Vertex B': Coordinates are [tex]\((-6, 8)\)[/tex].
[tex]\[ B'' = (-(-6), -(8)) = (6, -8) \][/tex]

- Vertex C': Coordinates are [tex]\((-10, 5)\)[/tex].
[tex]\[ C'' = (-(-10), -(5)) = (10, -5) \][/tex]

Thus, the vertices of [tex]\(\Delta A''B''C''\)[/tex] are:
[tex]\[ A''(6, -5), B''(6, -8), C''(10, -5) \][/tex]

### Congruence of Triangles

The translation moves the vertices but preserves distances and angles between them. Rotating 180 degrees about the origin preserves distances and angles as well, thus preserving the shape and size of the triangle.

Since these transformations do not alter the relative distances and angles between the vertices, we can conclude that:

[tex]\[ \Delta ABC \cong \Delta A'B'C' \cong \Delta A''B''C'' \][/tex]

Therefore, [tex]\(\Delta ABC\)[/tex] is congruent to [tex]\(\Delta A'B'C'\)[/tex], and both are congruent to [tex]\(\Delta A''B''C''\)[/tex].

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