Question

Imagine a counterclockwise rotation of 270° about the origin of the figure. Choose the coordinate of A’.

(2,-1)

(1,1)

(1,2)

(-1,-2)

Answer 1

To find the coordinates of point A' after a counterclockwise rotation of 270° about the origin, we can use the rotation formulas related to counterclockwise transformations.

The general formula for a counterclockwise rotation of θ degrees about the origin is:
- \( x' = x \cos(\theta) - y \sin(\theta) \)
- \( y' = x \sin(\theta) + y \cos(\theta) \)

For a 270° rotation, we have:
- \( \cos(270°) = 0 \)
- \( \sin(270°) = -1 \)

Thus, the formulas simplify to:
- \( x' = y \)
- \( y' = -x \)

Now, we will apply this to each of the given points to find the new coordinates after the rotation.

1. For **(2, -1)**:
- \( x' = -1 \)
- \( y' = -2 \)
- Result: **(-1, -2)**

2. For **(1, 1)**:
- \( x' = 1 \)
- \( y' = -1 \)
- Result: **(1, -1)**

3. For **(1, 2)**:
- \( x' = 2 \)
- \( y' = -1 \)
- Result: **(2, -1)**

4. For **(-1, -2)**:
- \( x' = -2 \)
- \( y' = 1 \)
- Result: **(-2, 1)**

After checking all the calculations from the given points, it’s clear that the point (2,-1) rotates to (-1, -2) (the other points do not match).

So, the new coordinates A' after a counterclockwise rotation of 270° about the origin for point \( (2, -1) \) is:

**A' = (-1, -2)**.