Triangle [tex]\(ABC\)[/tex] 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]\((x, y) \rightarrow (x-4, y+3)\)[/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.

3. Part C: Triangle [tex]\(\triangle A'B'C'\)[/tex] is rotated [tex]\(90^\circ\)[/tex] 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.



Answer :

Sure, let's solve the problem step by step.

### Part A: Translation Description

Given the translation rule [tex]\((x, y) \rightarrow (x - 4, y + 3)\)[/tex], we need to describe this in words.

Answer:
The translation can be described as "moving each point 4 units to the left and 3 units up."

### Part B: Determining the Vertices of [tex]\(\triangle A'B'C'\)[/tex]

To find the new coordinates of the vertices after translation, we apply the translation rule to each vertex of triangle [tex]\(ABC\)[/tex].

1. For vertex [tex]\(A(-3, 1)\)[/tex]:
[tex]\[ A' = (-3 - 4, 1 + 3) = (-7, 4) \][/tex]

2. For vertex [tex]\(B(-3, 4)\)[/tex]:
[tex]\[ B' = (-3 - 4, 4 + 3) = (-7, 7) \][/tex]

3. For vertex [tex]\(C(-7, 1)\)[/tex]:
[tex]\[ C' = (-7 - 4, 1 + 3) = (-11, 4) \][/tex]

Answer:
The vertices of [tex]\(\triangle A'B'C'\)[/tex] are located at:
- [tex]\(A'(-7, 4)\)[/tex]
- [tex]\(B'(-7, 7)\)[/tex]
- [tex]\(C'(-11, 4)\)[/tex]

### Part C: Rotation and Congruency

To rotate [tex]\(\triangle A'B'C'\)[/tex] 90 degrees clockwise about the origin, we use the rotation rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex].

1. For vertex [tex]\(A'(-7, 4)\)[/tex]:
[tex]\[ A'' = (4, -(-7)) = (4, 7) \][/tex]

2. For vertex [tex]\(B'(-7, 7)\)[/tex]:
[tex]\[ B'' = (7, -(-7)) = (7, 7) \][/tex]

3. For vertex [tex]\(C'(-11, 4)\)[/tex]:
[tex]\[ C'' = (4, -(-11)) = (4, 11) \][/tex]

Are [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle A'' B'' C''\)[/tex] congruent?

To check for congruency, we need to compare the side lengths of both triangles. The side lengths of [tex]\(\triangle ABC\)[/tex] are:

1. Side [tex]\(AB\)[/tex]:
[tex]\[ AB = \sqrt{((-3) - (-3))^2 + (4 - 1)^2} = \sqrt{0 + 3^2} = 3 \][/tex]

2. Side [tex]\(BC\)[/tex]:
[tex]\[ BC = \sqrt{((-7) - (-3))^2 + (1 - 4)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = 5 \][/tex]

3. Side [tex]\(CA\)[/tex]:
[tex]\[ CA = \sqrt{((-3) - (-7))^2 + (1 - 1)^2} = \sqrt{4^2 + 0} = 4 \][/tex]

The side lengths of [tex]\(\triangle A''B''C''\)[/tex] are the same as the side lengths of [tex]\(\triangle A'B'C'\)[/tex], since translation and rotation are rigid transformations (they don’t change side lengths).

Answer:
Since the side lengths of [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle A''B''C''\)[/tex] are the same ([tex]\(AB = A''B'' = 3\)[/tex], [tex]\(BC = B''C'' = 5\)[/tex], [tex]\(CA = C''A'' = 4\)[/tex]), we can conclude that:
[tex]\[ \triangle ABC \text{ is congruent to } \triangle A''B''C'' \][/tex]
Hence, [tex]\(\triangle A B C\)[/tex] is congruent to [tex]\(\triangle A'' B'' C''\)[/tex].