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].
