Answer :
Let's solve the problem step-by-step.
### Part A
To describe the translation in words, we need to interpret the given translation rule [tex]\((x, y) \rightarrow (x+4, y-3)\)[/tex]. This rule indicates that each vertex of the triangle is moved 4 units to the right and 3 units down. Therefore, the translation can be described as:
The triangle is translated 4 units to the right and 3 units down.
### Part B
To find the vertices of the translated triangle [tex]\( \triangle A^{\prime} B^{\prime} C^{\prime} \)[/tex], we apply the translation rule [tex]\((x, y) \rightarrow (x+4, y-3)\)[/tex] to each vertex of the original triangle [tex]\( \triangle ABC \)[/tex], where the original vertices are:
- [tex]\( A(-3, 1) \)[/tex]
- [tex]\( B(-3, 4) \)[/tex]
- [tex]\( C(-7, 1) \)[/tex]
Let's calculate the new coordinates:
1. For vertex [tex]\( A(-3, 1) \)[/tex]:
[tex]\[ A' = (-3 + 4, 1 - 3) = (1, -2) \][/tex]
2. For vertex [tex]\( B(-3, 4) \)[/tex]:
[tex]\[ B' = (-3 + 4, 4 - 3) = (1, 1) \][/tex]
3. For vertex [tex]\( C(-7, 1) \)[/tex]:
[tex]\[ C' = (-7 + 4, 1 - 3) = (-3, -2) \][/tex]
Therefore, the vertices of [tex]\( \triangle A^{\prime} B^{\prime} C^{\prime} \)[/tex] are:
- [tex]\( A' = (1, -2) \)[/tex]
- [tex]\( B' = (1, 1) \)[/tex]
- [tex]\( C' = (-3, -2) \)[/tex]
### Part C
To rotate [tex]\( \triangle A^{\prime} B^{\prime} C^{\prime} \)[/tex] 90 degrees counterclockwise about the origin and find the vertices of [tex]\( \triangle A^{\prime\prime} B^{\prime\prime} C^{\prime\prime} \)[/tex], we use the rotation rule for a point [tex]\((x, y) \)[/tex]: [tex]\((x, y) \rightarrow (-y, x)\)[/tex].
Applying this rule to each vertex of [tex]\( \triangle A^{\prime} B^{\prime} C^{\prime} \)[/tex]:
1. For vertex [tex]\( A' = (1, -2) \)[/tex]:
[tex]\[ A'' = (-(-2), 1) = (2, 1) \][/tex]
2. For vertex [tex]\( B' = (1, 1) \)[/tex]:
[tex]\[ B'' = (-(1), 1) = (-1, 1) \][/tex]
3. For vertex [tex]\( C' = (-3, -2) \)[/tex]:
[tex]\[ C'' = (-(-2), -3) = (2, -3) \][/tex]
Therefore, the vertices of [tex]\( \triangle A'' B'' C'' \)[/tex] are:
- [tex]\( A'' = (2, 1) \)[/tex]
- [tex]\( B'' = (-1, 1) \)[/tex]
- [tex]\( C'' = (2, -3) \)[/tex]
To determine if [tex]\( \triangle ABC \)[/tex] is congruent to [tex]\( \triangle A'B''C'' \)[/tex], we should consider that:
- Translation and rotation are rigid transformations that preserve the lengths of sides and magnitudes of angles.
- Because both transformations do not alter the size or shape of the triangle, the resulting triangle [tex]\( \triangle A'' B'' C'' \)[/tex] is congruent to the original triangle [tex]\( \triangle ABC \)[/tex].
Therefore, the triangles are indeed congruent.
In conclusion:
1. The translation can be described as "The triangle is translated 4 units to the right and 3 units down."
2. The vertices of the translated triangle [tex]\( \triangle A^{\prime} B^{\prime} C^{\prime} \)[/tex] are [tex]\( A' = (1, -2) \)[/tex], [tex]\( B' = (1, 1) \)[/tex], and [tex]\( C' = (-3, -2) \)[/tex].
3. After a 90-degree counterclockwise rotation, the vertices of the triangle [tex]\( \triangle A'' B'' C'' \)[/tex] are [tex]\( A'' = (2, 1) \)[/tex], [tex]\( B'' = (-1, 1) \)[/tex], and [tex]\( C'' = (2, -3) \)[/tex]. [tex]\( \triangle ABC \)[/tex] is congruent to [tex]\( \triangle A'' B'' C'' \)[/tex] because translation and rotation are rigid transformations that preserve the size and shape of the triangle.
### Part A
To describe the translation in words, we need to interpret the given translation rule [tex]\((x, y) \rightarrow (x+4, y-3)\)[/tex]. This rule indicates that each vertex of the triangle is moved 4 units to the right and 3 units down. Therefore, the translation can be described as:
The triangle is translated 4 units to the right and 3 units down.
### Part B
To find the vertices of the translated triangle [tex]\( \triangle A^{\prime} B^{\prime} C^{\prime} \)[/tex], we apply the translation rule [tex]\((x, y) \rightarrow (x+4, y-3)\)[/tex] to each vertex of the original triangle [tex]\( \triangle ABC \)[/tex], where the original vertices are:
- [tex]\( A(-3, 1) \)[/tex]
- [tex]\( B(-3, 4) \)[/tex]
- [tex]\( C(-7, 1) \)[/tex]
Let's calculate the new coordinates:
1. For vertex [tex]\( A(-3, 1) \)[/tex]:
[tex]\[ A' = (-3 + 4, 1 - 3) = (1, -2) \][/tex]
2. For vertex [tex]\( B(-3, 4) \)[/tex]:
[tex]\[ B' = (-3 + 4, 4 - 3) = (1, 1) \][/tex]
3. For vertex [tex]\( C(-7, 1) \)[/tex]:
[tex]\[ C' = (-7 + 4, 1 - 3) = (-3, -2) \][/tex]
Therefore, the vertices of [tex]\( \triangle A^{\prime} B^{\prime} C^{\prime} \)[/tex] are:
- [tex]\( A' = (1, -2) \)[/tex]
- [tex]\( B' = (1, 1) \)[/tex]
- [tex]\( C' = (-3, -2) \)[/tex]
### Part C
To rotate [tex]\( \triangle A^{\prime} B^{\prime} C^{\prime} \)[/tex] 90 degrees counterclockwise about the origin and find the vertices of [tex]\( \triangle A^{\prime\prime} B^{\prime\prime} C^{\prime\prime} \)[/tex], we use the rotation rule for a point [tex]\((x, y) \)[/tex]: [tex]\((x, y) \rightarrow (-y, x)\)[/tex].
Applying this rule to each vertex of [tex]\( \triangle A^{\prime} B^{\prime} C^{\prime} \)[/tex]:
1. For vertex [tex]\( A' = (1, -2) \)[/tex]:
[tex]\[ A'' = (-(-2), 1) = (2, 1) \][/tex]
2. For vertex [tex]\( B' = (1, 1) \)[/tex]:
[tex]\[ B'' = (-(1), 1) = (-1, 1) \][/tex]
3. For vertex [tex]\( C' = (-3, -2) \)[/tex]:
[tex]\[ C'' = (-(-2), -3) = (2, -3) \][/tex]
Therefore, the vertices of [tex]\( \triangle A'' B'' C'' \)[/tex] are:
- [tex]\( A'' = (2, 1) \)[/tex]
- [tex]\( B'' = (-1, 1) \)[/tex]
- [tex]\( C'' = (2, -3) \)[/tex]
To determine if [tex]\( \triangle ABC \)[/tex] is congruent to [tex]\( \triangle A'B''C'' \)[/tex], we should consider that:
- Translation and rotation are rigid transformations that preserve the lengths of sides and magnitudes of angles.
- Because both transformations do not alter the size or shape of the triangle, the resulting triangle [tex]\( \triangle A'' B'' C'' \)[/tex] is congruent to the original triangle [tex]\( \triangle ABC \)[/tex].
Therefore, the triangles are indeed congruent.
In conclusion:
1. The translation can be described as "The triangle is translated 4 units to the right and 3 units down."
2. The vertices of the translated triangle [tex]\( \triangle A^{\prime} B^{\prime} C^{\prime} \)[/tex] are [tex]\( A' = (1, -2) \)[/tex], [tex]\( B' = (1, 1) \)[/tex], and [tex]\( C' = (-3, -2) \)[/tex].
3. After a 90-degree counterclockwise rotation, the vertices of the triangle [tex]\( \triangle A'' B'' C'' \)[/tex] are [tex]\( A'' = (2, 1) \)[/tex], [tex]\( B'' = (-1, 1) \)[/tex], and [tex]\( C'' = (2, -3) \)[/tex]. [tex]\( \triangle ABC \)[/tex] is congruent to [tex]\( \triangle A'' B'' C'' \)[/tex] because translation and rotation are rigid transformations that preserve the size and shape of the triangle.
