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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.

What is the ordered pair of point C

9 answers

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 \((1, 3)\).

Point A (-3, 4) is rotated to become A' (-4, -3).
What degree of rotation is this? (Counterclockwise)

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.

The point A(1, 2) is rotated 270 degrees about the origin. (Counterclockwise)
What is the ordered pair for A'?

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 \( (2, -1) \).

A figure is rotated 180 degrees about the origin. Which of the following function mappings was applied?

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.

Provide two different degrees of rotation less than 120 degrees but greater than 0 that will turn a regular

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.