Question

Given a point ( x, y
), what is the new point when the point is rotated counterclockwise about the origin 270 degrees

Answer 1

Given a point on a geometric figure (1, 2)
, what is the new point when the point is rotated clockwise about the origin 90 degrees

Answer 2

To rotate a point \((x, y)\) counterclockwise about the origin by \(270\) degrees, you can use the rotation formula. The general formula for rotating a point \((x, y)\) counterclockwise by an angle \(\theta\) is given by:

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

For a rotation of \(270\) degrees, we convert this angle to radians:

\[
270 \text{ degrees} = \frac{270 \pi}{180} = \frac{3\pi}{2} \text{ radians}
\]

Next, we evaluate \(\cos(270^\circ)\) and \(\sin(270^\circ)\):

- \(\cos(270^\circ) = 0\)
- \(\sin(270^\circ) = -1\)

Substituting these values into the rotation formula:

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

This simplifies to:

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

So, the new point after rotating the point \((x, y)\) counterclockwise by \(270\) degrees about the origin is:

\[
(y, -x)
\]