Question
Triangle ABC is shown on the graph. What are the coordinates of the image of point B after the triangle is rotated 270° about the origin?
(4, 2)
(2, 4)
(–4, –2)
(–2, –4)
Answers
Answer
Point B is (-2,4)
Answered by
GPT-4o mini
To find the coordinates of point B after a 270° rotation about the origin, we can use the rotation transformation rules.
The general rule for rotating a point \((x, y)\) counterclockwise by 270° (which is the same as a clockwise rotation of 90°) about the origin is given by:
\[
(x, y) \rightarrow (y, -x)
\]
If we denote the original point \(B\) as \((x, y)\), after applying the transformation for a 270° rotation:
1. Let \(B\) be at coordinates \((x, y)\).
2. After the rotation, the new coordinates will be \((y, -x)\).
Now, without knowing the exact coordinates of point B, we cannot calculate the result directly. However, let's consider each of the answer choices provided and see how to reach them:
1. If \(B = (4, 2)\):
- After rotation: \((2, -4)\)
2. If \(B = (2, 4)\):
- After rotation: \((4, -2)\)
3. If \(B = (-4, -2)\):
- After rotation: \((-2, 4)\)
4. If \(B = (-2, -4)\):
- After rotation: \((-4, 2)\)
Among the options provided:
- \((4, 2)\) rotates to \((2, -4)\)
- \((2, 4)\) rotates to \((4, -2)\)
- \((-4, -2)\) rotates to \((-2, 4)\)
- \((-2, -4)\) rotates to \((-4, 2)\)
It's important to determine the starting coordinates of point B. If you check the coordinates of B directly from your graph, you can apply the rotations above accordingly.
Please provide the coordinates of point B, and I can give the exact answer based on the rotation.
The general rule for rotating a point \((x, y)\) counterclockwise by 270° (which is the same as a clockwise rotation of 90°) about the origin is given by:
\[
(x, y) \rightarrow (y, -x)
\]
If we denote the original point \(B\) as \((x, y)\), after applying the transformation for a 270° rotation:
1. Let \(B\) be at coordinates \((x, y)\).
2. After the rotation, the new coordinates will be \((y, -x)\).
Now, without knowing the exact coordinates of point B, we cannot calculate the result directly. However, let's consider each of the answer choices provided and see how to reach them:
1. If \(B = (4, 2)\):
- After rotation: \((2, -4)\)
2. If \(B = (2, 4)\):
- After rotation: \((4, -2)\)
3. If \(B = (-4, -2)\):
- After rotation: \((-2, 4)\)
4. If \(B = (-2, -4)\):
- After rotation: \((-4, 2)\)
Among the options provided:
- \((4, 2)\) rotates to \((2, -4)\)
- \((2, 4)\) rotates to \((4, -2)\)
- \((-4, -2)\) rotates to \((-2, 4)\)
- \((-2, -4)\) rotates to \((-4, 2)\)
It's important to determine the starting coordinates of point B. If you check the coordinates of B directly from your graph, you can apply the rotations above accordingly.
Please provide the coordinates of point B, and I can give the exact answer based on the rotation.
Answered by
GPT-4o mini
Given that the coordinates of point B are \((-2, 4)\), we will apply the 270° rotation transformation about the origin.
Using the transformation rule for a 270° rotation, which is:
\[
(x, y) \rightarrow (y, -x)
\]
we substitute the coordinates of point B:
- \(x = -2\)
- \(y = 4\)
Now applying the transformation:
\[
(-2, 4) \rightarrow (4, -(-2)) \rightarrow (4, 2)
\]
After the rotation, the coordinates of point B will be \((4, 2)\).
Thus, the coordinates of the image of point B after the triangle is rotated 270° about the origin are \(\boxed{(4, 2)}\).
Using the transformation rule for a 270° rotation, which is:
\[
(x, y) \rightarrow (y, -x)
\]
we substitute the coordinates of point B:
- \(x = -2\)
- \(y = 4\)
Now applying the transformation:
\[
(-2, 4) \rightarrow (4, -(-2)) \rightarrow (4, 2)
\]
After the rotation, the coordinates of point B will be \((4, 2)\).
Thus, the coordinates of the image of point B after the triangle is rotated 270° about the origin are \(\boxed{(4, 2)}\).
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