Question
Figure
\[P\] was rotated about the origin
\[(0,0)\] by
\[90^\circ\] counterclockwise.
A coordinate plane with 1 triangle. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[P\]
Which figure is the image of
\[P\]?
Choose 1 answer:
Choose 1 answer:
(Choice A) A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle A. Triangle A has vertices at (2, negative 6), (2, negative 2), and (7, negative 3). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[A\]
\[P\]
A
A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle A. Triangle A has vertices at (2, negative 6), (2, negative 2), and (7, negative 3). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[A\]
\[P\]
(Choice B) A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle B. Triangle B has vertices at (3, negative 5), (3, negative 1), and (8, negative 2). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[B\]
\[P\]
B
A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle B. Triangle B has vertices at (3, negative 5), (3, negative 1), and (8, negative 2). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[B\]
\[P\]
(Choice C) A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle C. Triangle C has vertices at (2, negative 2), (6, negative 2), and (3, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[C\]
\[P\]
C
A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle C. Triangle C has vertices at (2, negative 2), (6, negative 2), and (3, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[C\]
\[P\]
(Choice D) A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle D. Triangle D has vertices at (2, negative 2), (6, negative 2), and (5, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[D\]
\[P\]
D
A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle D. Triangle D has vertices at (2, negative 2), (6, negative 2), and (5, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[D\]
\[P\]
\[P\] was rotated about the origin
\[(0,0)\] by
\[90^\circ\] counterclockwise.
A coordinate plane with 1 triangle. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[P\]
Which figure is the image of
\[P\]?
Choose 1 answer:
Choose 1 answer:
(Choice A) A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle A. Triangle A has vertices at (2, negative 6), (2, negative 2), and (7, negative 3). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[A\]
\[P\]
A
A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle A. Triangle A has vertices at (2, negative 6), (2, negative 2), and (7, negative 3). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[A\]
\[P\]
(Choice B) A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle B. Triangle B has vertices at (3, negative 5), (3, negative 1), and (8, negative 2). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[B\]
\[P\]
B
A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle B. Triangle B has vertices at (3, negative 5), (3, negative 1), and (8, negative 2). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[B\]
\[P\]
(Choice C) A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle C. Triangle C has vertices at (2, negative 2), (6, negative 2), and (3, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[C\]
\[P\]
C
A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle C. Triangle C has vertices at (2, negative 2), (6, negative 2), and (3, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[C\]
\[P\]
(Choice D) A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle D. Triangle D has vertices at (2, negative 2), (6, negative 2), and (5, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[D\]
\[P\]
D
A coordinate plane with 2 triangles. Triangle P has vertices at (negative 6, negative 2), (negative 2, negative 2), and (negative 3, negative 7). Triangle P is rotated to form triangle D. Triangle D has vertices at (2, negative 2), (6, negative 2), and (5, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[D\]
\[P\]
Answers
Answered by
GPT-4o mini
To find the image of triangle \( P \) after a \( 90^\circ \) counterclockwise rotation around the origin \( (0,0) \), we can use the following transformation rule:
If a point has coordinates \( (x, y) \), after a \( 90^\circ \) counterclockwise rotation, its new coordinates will be given by:
\[
(x', y') = (-y, x)
\]
Now, let's apply this transformation to the vertices of triangle \( P \):
1. For the vertex \( (-6, -2) \):
\[
x' = -(-2) = 2, \quad y' = -6 \quad \Rightarrow \quad (2, -6)
\]
2. For the vertex \( (-2, -2) \):
\[
x' = -(-2) = 2, \quad y' = -2 \quad \Rightarrow \quad (2, -2)
\]
3. For the vertex \( (-3, -7) \):
\[
x' = -(-7) = 7, \quad y' = -3 \quad \Rightarrow \quad (7, -3)
\]
Thus, after the rotation, the vertices of the rotated triangle (let's call it triangle \( A \)) will be at:
- \( (2, -6) \)
- \( (2, -2) \)
- \( (7, -3) \)
Now, let's compare these coordinates to the options provided:
- **Choice A**: Triangle A has vertices at \( (2, -6) \), \( (2, -2) \), and \( (7, -3) \). This matches our calculated coordinates.
- **Choice B**: Vertices are \( (3, -5) \), \( (3, -1) \), and \( (8, -2) \). Doesn't match.
- **Choice C**: Vertices are \( (2, -2) \), \( (6, -2) \), and \( (3, -7) \). Doesn't match.
- **Choice D**: Vertices are \( (2, -2) \), \( (6, -2) \), and \( (5, -7) \). Doesn't match.
From the analysis, the correct image of \( P \) after a \( 90^\circ \) counterclockwise rotation is **Choice A**.
If a point has coordinates \( (x, y) \), after a \( 90^\circ \) counterclockwise rotation, its new coordinates will be given by:
\[
(x', y') = (-y, x)
\]
Now, let's apply this transformation to the vertices of triangle \( P \):
1. For the vertex \( (-6, -2) \):
\[
x' = -(-2) = 2, \quad y' = -6 \quad \Rightarrow \quad (2, -6)
\]
2. For the vertex \( (-2, -2) \):
\[
x' = -(-2) = 2, \quad y' = -2 \quad \Rightarrow \quad (2, -2)
\]
3. For the vertex \( (-3, -7) \):
\[
x' = -(-7) = 7, \quad y' = -3 \quad \Rightarrow \quad (7, -3)
\]
Thus, after the rotation, the vertices of the rotated triangle (let's call it triangle \( A \)) will be at:
- \( (2, -6) \)
- \( (2, -2) \)
- \( (7, -3) \)
Now, let's compare these coordinates to the options provided:
- **Choice A**: Triangle A has vertices at \( (2, -6) \), \( (2, -2) \), and \( (7, -3) \). This matches our calculated coordinates.
- **Choice B**: Vertices are \( (3, -5) \), \( (3, -1) \), and \( (8, -2) \). Doesn't match.
- **Choice C**: Vertices are \( (2, -2) \), \( (6, -2) \), and \( (3, -7) \). Doesn't match.
- **Choice D**: Vertices are \( (2, -2) \), \( (6, -2) \), and \( (5, -7) \). Doesn't match.
From the analysis, the correct image of \( P \) after a \( 90^\circ \) counterclockwise rotation is **Choice A**.