Question
Which transformations can be used to map a triangle with vertices A(2, 2), B(4, 1), C(4, 5) to A’(–2, –2), B’(–1, –4), C’(–5, –4)?
a 180degree rotation about the origin
a 90degree counterclockwise rotation about the origin and a translation down 4 units
a 90degree clockwise rotation about the origin and a reflection over the y-axis
a reflection over the y-axis and then a 90degree clockwise rotation about the origin option 1 2 3 or 4
a 180degree rotation about the origin
a 90degree counterclockwise rotation about the origin and a translation down 4 units
a 90degree clockwise rotation about the origin and a reflection over the y-axis
a reflection over the y-axis and then a 90degree clockwise rotation about the origin option 1 2 3 or 4
Answers
GPT-4o mini
To determine which set of transformations can map triangle ABC to triangle A'B'C', we need to analyze the vertices and how they relate to each transformation option.
1. **Triangle ABC Vertices**:
A(2, 2)
B(4, 1)
C(4, 5)
2. **Triangle A'B'C' Vertices**:
A’(–2, –2)
B’(–1, –4)
C’(–5, –4)
Let's evaluate each option one by one:
### Option 1: 180-degree rotation about the origin
A 180-degree rotation about the origin will change the coordinates (x, y) to (-x, -y).
- A(2, 2) rotates to A'(-2, -2)
- B(4, 1) rotates to B'(-4, -1)
- C(4, 5) rotates to C'(-4, -5)
This does not match B’(–1, –4) and C’(–5, –4), so this option is incorrect.
### Option 2: 90-degree counterclockwise rotation about the origin and a translation down 4 units
A 90-degree counterclockwise rotation changes (x, y) to (-y, x). Let's apply that to each vertex:
- A(2, 2) becomes A'(-2, 2)
- B(4, 1) becomes B'(-1, 4)
- C(4, 5) becomes C'(-5, 4)
Then, we would translate down 4 units, changing (x, y) to (x, y - 4):
- A'(-2, 2) translates to (-2, -2)
- B'(-1, 4) translates to (-1, 0)
- C'(-5, 4) translates to (-5, 0)
Both B'(-1, 0) does not match B’(–1, –4) and C'(-5, 0) does not match C’(–5, –4), so this option is incorrect.
### Option 3: 90-degree clockwise rotation about the origin and a reflection over the y-axis
A 90-degree clockwise rotation changes (x, y) to (y, -x):
- A(2, 2) becomes A'(2, -2)
- B(4, 1) becomes B'(1, -4)
- C(4, 5) becomes C'(-5, -4)
After this step if we now reflect over the y-axis, (x, y) becomes (-x, y):
- A'(2, -2) reflects to A'(-2, -2)
- B'(1, -4) reflects to B'(-1, -4)
- C'(-5, -4) remains C'(-5, -4)
This matches A’(–2, –2), B’(–1, –4), and C’(–5, –4). Thus, this option works.
### Option 4: Reflection over the y-axis and then a 90-degree clockwise rotation about the origin
The reflection over the y-axis would change (x, y) to (-x, y):
- A(2, 2) becomes A'(-2, 2)
- B(4, 1) becomes B'(-4, 1)
- C(4, 5) becomes C'(-4, 5)
Then we would perform the 90-degree clockwise rotation:
- A'(-2, 2) becomes A'(2, 2)
- B'(-4, 1) becomes B'(1, 4)
- C'(-4, 5) becomes C'(-5, 4)
B' and C' do not match the target triangles, so this option is incorrect.
In conclusion, the only transformation that correctly maps triangle ABC to triangle A'B'C' is **Option 3**: a 90-degree clockwise rotation about the origin and a reflection over the y-axis.
1. **Triangle ABC Vertices**:
A(2, 2)
B(4, 1)
C(4, 5)
2. **Triangle A'B'C' Vertices**:
A’(–2, –2)
B’(–1, –4)
C’(–5, –4)
Let's evaluate each option one by one:
### Option 1: 180-degree rotation about the origin
A 180-degree rotation about the origin will change the coordinates (x, y) to (-x, -y).
- A(2, 2) rotates to A'(-2, -2)
- B(4, 1) rotates to B'(-4, -1)
- C(4, 5) rotates to C'(-4, -5)
This does not match B’(–1, –4) and C’(–5, –4), so this option is incorrect.
### Option 2: 90-degree counterclockwise rotation about the origin and a translation down 4 units
A 90-degree counterclockwise rotation changes (x, y) to (-y, x). Let's apply that to each vertex:
- A(2, 2) becomes A'(-2, 2)
- B(4, 1) becomes B'(-1, 4)
- C(4, 5) becomes C'(-5, 4)
Then, we would translate down 4 units, changing (x, y) to (x, y - 4):
- A'(-2, 2) translates to (-2, -2)
- B'(-1, 4) translates to (-1, 0)
- C'(-5, 4) translates to (-5, 0)
Both B'(-1, 0) does not match B’(–1, –4) and C'(-5, 0) does not match C’(–5, –4), so this option is incorrect.
### Option 3: 90-degree clockwise rotation about the origin and a reflection over the y-axis
A 90-degree clockwise rotation changes (x, y) to (y, -x):
- A(2, 2) becomes A'(2, -2)
- B(4, 1) becomes B'(1, -4)
- C(4, 5) becomes C'(-5, -4)
After this step if we now reflect over the y-axis, (x, y) becomes (-x, y):
- A'(2, -2) reflects to A'(-2, -2)
- B'(1, -4) reflects to B'(-1, -4)
- C'(-5, -4) remains C'(-5, -4)
This matches A’(–2, –2), B’(–1, –4), and C’(–5, –4). Thus, this option works.
### Option 4: Reflection over the y-axis and then a 90-degree clockwise rotation about the origin
The reflection over the y-axis would change (x, y) to (-x, y):
- A(2, 2) becomes A'(-2, 2)
- B(4, 1) becomes B'(-4, 1)
- C(4, 5) becomes C'(-4, 5)
Then we would perform the 90-degree clockwise rotation:
- A'(-2, 2) becomes A'(2, 2)
- B'(-4, 1) becomes B'(1, 4)
- C'(-4, 5) becomes C'(-5, 4)
B' and C' do not match the target triangles, so this option is incorrect.
In conclusion, the only transformation that correctly maps triangle ABC to triangle A'B'C' is **Option 3**: a 90-degree clockwise rotation about the origin and a reflection over the y-axis.