Question

Imagine a counterclockwise rotation of 270° about the origin of the figure. Choose the coordinate of A′.
Answer
The cordinates for a are (-2, 1)
Answer
Soory the cordinates are actually (1, -2)
Answered by GPT-4o mini
To find the coordinates of point A' after a counterclockwise rotation of 270° about the origin, we can use the rotation transformation formulas.

The general formula for a point \( (x, y) \) rotated about the origin by an angle \( \theta \) is given by:
\[
(x', y') = (x \cos(\theta) - y \sin(\theta), x \sin(\theta) + y \cos(\theta))
\]

For a 270° rotation counterclockwise, \( \theta = 270° \). We can convert degrees to radians since many trigonometric functions use radians. The equivalent angle in radians is:
\[
\theta = 270° = \frac{3\pi}{2} \text{ radians}
\]

Now, we calculate the cosine and sine of 270°:
\[
\cos(270°) = 0
\]
\[
\sin(270°) = -1
\]

So the transformation equations for a 270° rotation are:
\[
(x', y') = (x \cdot 0 - y \cdot (-1), x \cdot (-1) + y \cdot 0)
\]
This simplifies to:
\[
(x', y') = (y, -x)
\]

To find the coordinates of \( A' \), we need the original coordinates of point A. If you provide the coordinates of point A (let's say \( A(x, y) \)), we can substitute those values into the equations and compute \( A' \).

For example, if the coordinates of A are (2, 3):
\[
A' = (3, -2)
\]

Please provide the coordinates of point A for a specific answer!
Answered by GPT-4o mini
To find the coordinates of point \( A' \) after a 270° counterclockwise rotation about the origin for the original point \( A(-2, 1) \), we use the transformation we derived earlier:

The transformation for a 270° rotation is:
\[
(x', y') = (y, -x)
\]

Given the coordinates of \( A \) are \( (-2, 1) \):
- \( x = -2 \)
- \( y = 1 \)

Now we can apply the transformation:
\[
x' = y = 1
\]
\[
y' = -x = -(-2) = 2
\]

Thus, the coordinates of point \( A' \) after the rotation are:
\[
A' = (1, 2)
\]

So, the final answer is:
**The coordinates of \( A' \) are \( (1, 2) \).**
Answered by GPT-4o mini
No problem! Let's recalculate the coordinates \( A' \) after a 270° counterclockwise rotation about the origin, using the correct original point \( A(1, -2) \).

Using the transformation for a 270° rotation:
\[
(x', y') = (y, -x)
\]

Given:
- \( x = 1 \)
- \( y = -2 \)

Now we substitute these values into the transformation:
\[
x' = y = -2
\]
\[
y' = -x = -1
\]

Thus, the coordinates of point \( A' \) after the rotation are:
\[
A' = (-2, -1)
\]

So, the final answer is:
**The coordinates of \( A' \) are \( (-2, -1) \).**