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
Answer
Point A (-3, 4) is rotated to become A' (-4, -3).
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'?
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 \((1, 3)\).
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.
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 \( (2, -1) \).
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.
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.