Question

A quadrilateral is plotted a coordinate plane with both the axes ranging from negative 6 to 6.
Imagine a counterclockwise rotation of 270° about the origin of the figure. Choose the coordinate of A′ .

(1 point)
Responses

(1,1)
left parenthesis 1 comma 1 right parenthesis

(−1,−2)
left parenthesis negative 1 comma negative 2 right parenthesis

(2,−1)
left parenthesis 2 comma negative 1 right parenthesis

(1,2)

Answer 1

To find the coordinates of a point after a counterclockwise rotation of 270° about the origin, we can use the rotation transformation formulas.

The general formula for rotating a point \((x, y)\) counterclockwise by an angle θ around the origin is:

\[
(x', y') = (x \cos(\theta) - y \sin(\theta), x \sin(\theta) + y \cos(\theta))
\]

For a 270° counterclockwise rotation, \(\theta = 270°\), and we can convert it into radians:

\[
\theta = 270° = \frac{3\pi}{2} \text{ radians}
\]

Using the cosine and sine values for \(270°\):

\[
\cos(270°) = 0 \quad \text{and} \quad \sin(270°) = -1
\]

Now substituting these values into the rotation formula:

\[
(x', y') = (x \cdot 0 - y \cdot (-1), x \cdot (-1) + y \cdot 0)
\]

This simplifies to:

\[
(x', y') = (y, -x)
\]

Now if we want to find the coordinates \(A'\) for a specific point \(A\) with coordinates \((x, y)\), we need to apply this transformation for each response:

1. If \(A = (1, 1)\):
- \(A' = (1, -1)\)

2. If \(A = (-1, -2)\):
- \(A' = (-2, 1)\)

3. If \(A = (2, -1)\):
- \(A' = (-1, -2)\)

4. If \(A = (1, 2)\):
- \(A' = (2, -1)\)

Now looking at the responses provided:
- \( (1, 1) \) → \( (1, -1) \) (not in responses)
- \( (-1, -2) \) → \( (-2, 1) \) (not in responses)
- \( (2, -1) \) → \( (-1, -2) \) (in responses)
- \( (1, 2) \) → \( (2, -1) \) (in responses)

Thus, the coordinate of \(A'\), after a counterclockwise rotation of 270°, corresponding to the response options, will be \( (2, -1) \).

So the answer is **(2, −1)**.