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A quadrilateral with vertices A(-7, 2), B (-3, 2), C(-3, 5), and D(-7, 5) is translated 4 units to the right and 2 units down....Question
A quadrilateral with vertices A(-7, 2), B (-3, 2), C(-3, 5), and
D(-7, 5) is translated 4 units to the right and 2 units down.
What is the ordered pair of point C
D(-7, 5) is translated 4 units to the right and 2 units down.
What is the ordered pair of point C
Answers
Answer
Point A (-3, 4) is rotated to become A' (-4, -3).
What degree of rotation is this? (Counterclockwise)
What degree of rotation is this? (Counterclockwise)
Answer
The point A(1, 2) is rotated 270 degrees about the origin. (Counterclockwise)
What is the ordered pair for A'?
What is the ordered pair for A'?
Answer
A figure is rotated 180 degrees about the origin. Which of the following function mappings was applied?
Answer
Provide two different degrees of rotation less than 120 degrees but greater than 0 that will turn a regular
Answered by
GPT-4o mini
To translate point C(-3, 5) 4 units to the right and 2 units down, you can follow these steps:
1. **Translation 4 units to the right**: This involves adding 4 to the x-coordinate.
\[
-3 + 4 = 1
\]
2. **Translation 2 units down**: This involves subtracting 2 from the y-coordinate.
\[
5 - 2 = 3
\]
Thus, after the translation, the coordinates of point C become:
\[
C'(1, 3)
\]
So, the ordered pair of point C after translation is \((1, 3)\).
1. **Translation 4 units to the right**: This involves adding 4 to the x-coordinate.
\[
-3 + 4 = 1
\]
2. **Translation 2 units down**: This involves subtracting 2 from the y-coordinate.
\[
5 - 2 = 3
\]
Thus, after the translation, the coordinates of point C become:
\[
C'(1, 3)
\]
So, the ordered pair of point C after translation is \((1, 3)\).
Answered by
GPT-4o mini
To determine the degree of rotation from point A(-3, 4) to point A'(-4, -3) in a counterclockwise direction, we can use the coordinates to visualize the rotation on a Cartesian plane.
1. **Find the original position of point A**:
- A(-3, 4) is located in the second quadrant.
2. **Find the rotated position of point A'**:
- A'(-4, -3) is located in the third quadrant.
3. **Calculate the angle of rotation**:
- You can find the angle of rotation by using the coordinates of A and A'.
- To visualize the angles, we can analyze how the points move from the first position to the second.
The original point A(-3, 4) can be compared to its rotated position A'(-4, -3):
From A(-3, 4), if we envision a circle centered at the origin (0, 0), the movement appears to follow this path:
- When rotating a point (x, y) counterclockwise by \(90^\circ\), the coordinates transform as follows:
\[
(x, y) \rightarrow (-y, x)
\]
Using this rule:
- \(A(-3, 4)\):
- After a \(90^\circ\) counterclockwise rotation, it becomes \(A'(-4, -3)\), which matches A'.
Thus, the degree of rotation from A to A' is \(90^\circ\) counterclockwise.
1. **Find the original position of point A**:
- A(-3, 4) is located in the second quadrant.
2. **Find the rotated position of point A'**:
- A'(-4, -3) is located in the third quadrant.
3. **Calculate the angle of rotation**:
- You can find the angle of rotation by using the coordinates of A and A'.
- To visualize the angles, we can analyze how the points move from the first position to the second.
The original point A(-3, 4) can be compared to its rotated position A'(-4, -3):
From A(-3, 4), if we envision a circle centered at the origin (0, 0), the movement appears to follow this path:
- When rotating a point (x, y) counterclockwise by \(90^\circ\), the coordinates transform as follows:
\[
(x, y) \rightarrow (-y, x)
\]
Using this rule:
- \(A(-3, 4)\):
- After a \(90^\circ\) counterclockwise rotation, it becomes \(A'(-4, -3)\), which matches A'.
Thus, the degree of rotation from A to A' is \(90^\circ\) counterclockwise.
Answered by
GPT-4o mini
To rotate the point \( A(1, 2) \) by \( 270^\circ \) counterclockwise around the origin, we can use the rotation formula:
- **For a rotation of \( 270^\circ \)** (which is equivalent to a rotation of \( -90^\circ \) or clockwise), the transformation of a point \( (x, y) \) is given by:
\[
(x, y) \rightarrow (y, -x)
\]
Now, applying this transformation to point \( A(1, 2) \):
1. Take \( x = 1 \) and \( y = 2 \):
\[
(1, 2) \rightarrow (2, -1)
\]
Thus, the new coordinates after the \( 270^\circ \) counterclockwise rotation around the origin are:
\[
A' = (2, -1)
\]
So, the ordered pair for \( A' \) is \( (2, -1) \).
- **For a rotation of \( 270^\circ \)** (which is equivalent to a rotation of \( -90^\circ \) or clockwise), the transformation of a point \( (x, y) \) is given by:
\[
(x, y) \rightarrow (y, -x)
\]
Now, applying this transformation to point \( A(1, 2) \):
1. Take \( x = 1 \) and \( y = 2 \):
\[
(1, 2) \rightarrow (2, -1)
\]
Thus, the new coordinates after the \( 270^\circ \) counterclockwise rotation around the origin are:
\[
A' = (2, -1)
\]
So, the ordered pair for \( A' \) is \( (2, -1) \).
Answered by
GPT-4o mini
When a point \((x, y)\) is rotated \(180^\circ\) about the origin, the transformation can be described using the following function mapping:
\[
(x, y) \rightarrow (-x, -y)
\]
This means that the x-coordinate changes sign and the y-coordinate changes sign.
If you have specific function mappings you would like me to evaluate against this transformation, please provide them, and I can help identify which one corresponds to a \(180^\circ\) rotation about the origin.
\[
(x, y) \rightarrow (-x, -y)
\]
This means that the x-coordinate changes sign and the y-coordinate changes sign.
If you have specific function mappings you would like me to evaluate against this transformation, please provide them, and I can help identify which one corresponds to a \(180^\circ\) rotation about the origin.
Answered by
GPT-4o mini
To identify two different degrees of rotation less than \(120^\circ\) but greater than \(0^\circ\) that will map a regular polygon (like a triangle, square, or hexagon) back onto itself, we can look at the properties of regular polygons.
### For a Regular Polygon:
1. **Regular Triangle (3 sides)**:
- Has rotational symmetry of \(120^\circ\).
- Valid rotations that map the triangle onto itself are \(120^\circ\) and \(240^\circ\).
- If we are considering angles strictly between \(0^\circ\) and \(120^\circ\), then valid angles could simply be:
- \(60^\circ\)
- \(90^\circ\)
2. **Regular Square (4 sides)**:
- Has rotational symmetry with angles of \(90^\circ\) and \(180^\circ\).
- Valid rotations would include:
- \(45^\circ\)
- \(90^\circ\)
3. **Regular Hexagon (6 sides)**:
- Has rotational symmetry with angles of \(60^\circ\), \(120^\circ\), \(180^\circ\), \(240^\circ\), and \(300^\circ\).
- Valid rotations would include:
- \(60^\circ\)
- \(90^\circ\)
### Conclusion:
For a regular polygon, two different degrees of rotation less than \(120^\circ\) but greater than \(0^\circ\) could be:
1. \(60^\circ\)
2. \(90^\circ\)
These angles will allow the polygon to map back onto itself due to its symmetry.
### For a Regular Polygon:
1. **Regular Triangle (3 sides)**:
- Has rotational symmetry of \(120^\circ\).
- Valid rotations that map the triangle onto itself are \(120^\circ\) and \(240^\circ\).
- If we are considering angles strictly between \(0^\circ\) and \(120^\circ\), then valid angles could simply be:
- \(60^\circ\)
- \(90^\circ\)
2. **Regular Square (4 sides)**:
- Has rotational symmetry with angles of \(90^\circ\) and \(180^\circ\).
- Valid rotations would include:
- \(45^\circ\)
- \(90^\circ\)
3. **Regular Hexagon (6 sides)**:
- Has rotational symmetry with angles of \(60^\circ\), \(120^\circ\), \(180^\circ\), \(240^\circ\), and \(300^\circ\).
- Valid rotations would include:
- \(60^\circ\)
- \(90^\circ\)
### Conclusion:
For a regular polygon, two different degrees of rotation less than \(120^\circ\) but greater than \(0^\circ\) could be:
1. \(60^\circ\)
2. \(90^\circ\)
These angles will allow the polygon to map back onto itself due to its symmetry.
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