Question
Figure
\[Q\] was rotated about the origin
\[(0,0)\] by
\[270^\circ\] counterclockwise.
A coordinate plane with 1 quadrilateral. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[Q\]
Which figure is the image of
\[Q\]?
Choose 1 answer:
Choose 1 answer:
(Choice A) A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral A. Quadrilateral A has vertices at (negative 7, 3), (negative 1, 0), (3, 2), and (negative 5, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[A\]
\[Q\]
A
A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral A. Quadrilateral A has vertices at (negative 7, 3), (negative 1, 0), (3, 2), and (negative 5, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[A\]
\[Q\]
(Choice B) A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral B. Quadrilateral B has vertices at (negative 2, negative 3), (1, 0), (negative 3, 7), and (7, 5). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[B\]
\[Q\]
B
A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral B. Quadrilateral B has vertices at (negative 2, negative 3), (1, 0), (negative 3, 7), and (7, 5). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[B\]
\[Q\]
(Choice C) A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral C. Quadrilateral C has vertices at (2, negative 3), (1, 0), (3, 7), and (negative 7, 5). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[C\]
\[Q\]
C
A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral C. Quadrilateral C has vertices at (2, negative 3), (1, 0), (3, 7), and (negative 7, 5). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[C\]
\[Q\]
(Choice D) A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral D. Quadrilateral D has vertices at (negative 2, 3), (0, negative 1), (negative 3, negative 7), and (7, negative 5). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[D\]
\[Q\]
D
A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral D. Quadrilateral D has vertices at (negative 2, 3), (0, negative 1), (negative 3, negative 7), and (7, negative 5). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[D\]
\[Q\]
\[Q\] was rotated about the origin
\[(0,0)\] by
\[270^\circ\] counterclockwise.
A coordinate plane with 1 quadrilateral. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[Q\]
Which figure is the image of
\[Q\]?
Choose 1 answer:
Choose 1 answer:
(Choice A) A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral A. Quadrilateral A has vertices at (negative 7, 3), (negative 1, 0), (3, 2), and (negative 5, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[A\]
\[Q\]
A
A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral A. Quadrilateral A has vertices at (negative 7, 3), (negative 1, 0), (3, 2), and (negative 5, negative 7). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[A\]
\[Q\]
(Choice B) A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral B. Quadrilateral B has vertices at (negative 2, negative 3), (1, 0), (negative 3, 7), and (7, 5). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[B\]
\[Q\]
B
A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral B. Quadrilateral B has vertices at (negative 2, negative 3), (1, 0), (negative 3, 7), and (7, 5). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[B\]
\[Q\]
(Choice C) A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral C. Quadrilateral C has vertices at (2, negative 3), (1, 0), (3, 7), and (negative 7, 5). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[C\]
\[Q\]
C
A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral C. Quadrilateral C has vertices at (2, negative 3), (1, 0), (3, 7), and (negative 7, 5). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[C\]
\[Q\]
(Choice D) A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral D. Quadrilateral D has vertices at (negative 2, 3), (0, negative 1), (negative 3, negative 7), and (7, negative 5). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[D\]
\[Q\]
D
A coordinate plane with 2 quadrilaterals. Quadrilateral Q has vertices at (5, 7), (7, negative 3), (1, 0), and (negative 3, negative 2). Quadrilateral Q is rotated to form quadrilateral D. Quadrilateral D has vertices at (negative 2, 3), (0, negative 1), (negative 3, negative 7), and (7, negative 5). The x and y axes scale by one.
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[\small{2}\]
\[\small{4}\]
\[\small{6}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}6}\]
\[D\]
\[Q\]
Answers
Answered by
GPT-4o mini
To find the image of quadrilateral \(Q\) after it has been rotated \(270^\circ\) counterclockwise about the origin, we apply the rotation transformation to each vertex of quadrilateral \(Q\).
The general rule for rotation \(270^\circ\) counterclockwise is:
- \((x, y) \rightarrow (y, -x)\)
Now we will apply this transformation to each of the vertices of \(Q\):
1. For point \((5, 7)\):
\[
(5, 7) \rightarrow (7, -5)
\]
2. For point \((7, -3)\):
\[
(7, -3) \rightarrow (-3, -7)
\]
3. For point \((1, 0)\):
\[
(1, 0) \rightarrow (0, -1)
\]
4. For point \((-3, -2)\):
\[
(-3, -2) \rightarrow (-2, 3)
\]
Thus, the vertices of the rotated quadrilateral \(A\) are:
- \((7, -5)\)
- \((-3, -7)\)
- \((0, -1)\)
- \((-2, 3)\)
Next, we can compare these vertices with the options provided:
**Choice A**: Contains vertices \((-7, 3)\), \((-1, 0)\), \((3, 2)\), \((-5, -7)\)
**Choice B**: Contains vertices \((-2, -3)\), \((1, 0)\), \((-3, 7)\), \((7, 5)\)
**Choice C**: Contains vertices \((2, -3)\), \((1, 0)\), \((3, 7)\), \((-7, 5)\)
**Choice D**: Contains vertices \((-2, 3)\), \((0, -1)\), \((-3, -7)\), \((7, -5)\)
From our calculations, the correct answer matches **Choice D** with the vertices:
- \((-2, 3)\)
- \((0, -1)\)
- \((-3, -7)\)
- \((7, -5)\)
Thus, the image of \(Q\) after a \(270^\circ\) counterclockwise rotation about the origin is given by:
**Choice D**.
The general rule for rotation \(270^\circ\) counterclockwise is:
- \((x, y) \rightarrow (y, -x)\)
Now we will apply this transformation to each of the vertices of \(Q\):
1. For point \((5, 7)\):
\[
(5, 7) \rightarrow (7, -5)
\]
2. For point \((7, -3)\):
\[
(7, -3) \rightarrow (-3, -7)
\]
3. For point \((1, 0)\):
\[
(1, 0) \rightarrow (0, -1)
\]
4. For point \((-3, -2)\):
\[
(-3, -2) \rightarrow (-2, 3)
\]
Thus, the vertices of the rotated quadrilateral \(A\) are:
- \((7, -5)\)
- \((-3, -7)\)
- \((0, -1)\)
- \((-2, 3)\)
Next, we can compare these vertices with the options provided:
**Choice A**: Contains vertices \((-7, 3)\), \((-1, 0)\), \((3, 2)\), \((-5, -7)\)
**Choice B**: Contains vertices \((-2, -3)\), \((1, 0)\), \((-3, 7)\), \((7, 5)\)
**Choice C**: Contains vertices \((2, -3)\), \((1, 0)\), \((3, 7)\), \((-7, 5)\)
**Choice D**: Contains vertices \((-2, 3)\), \((0, -1)\), \((-3, -7)\), \((7, -5)\)
From our calculations, the correct answer matches **Choice D** with the vertices:
- \((-2, 3)\)
- \((0, -1)\)
- \((-3, -7)\)
- \((7, -5)\)
Thus, the image of \(Q\) after a \(270^\circ\) counterclockwise rotation about the origin is given by:
**Choice D**.