Question

Given a point on a geometric figure (x, y)
, what is the new point when the figure is rotated clockwise about the origin 270 degrees?(1 point)

Answer 1

To rotate a point \((x, y)\) clockwise about the origin by 270 degrees, we can use the rotation transformation formulas.

A clockwise rotation of 270 degrees is equivalent to a counterclockwise rotation of 90 degrees. The transformation formulas for a counterclockwise rotation by an angle \(\theta\) are given by:

\[
\begin{pmatrix}
x' \\
y'
\end{pmatrix}
=
\begin{pmatrix}
x \cos \theta - y \sin \theta \\
x \sin \theta + y \cos \theta
\end{pmatrix}
\]

For a 90-degree counterclockwise rotation, \(\theta = 90^\circ\) gives us:

\(\cos(90^\circ) = 0\)
\(\sin(90^\circ) = 1\)

Applying these values to the transformation:

\[
\begin{pmatrix}
x' \\
y'
\end{pmatrix}
=
\begin{pmatrix}
x \cdot 0 - y \cdot 1 \\
x \cdot 1 + y \cdot 0
\end{pmatrix}
=
\begin{pmatrix}
-y \\
x
\end{pmatrix}
\]

Therefore, to obtain the new point after a 270-degree clockwise rotation, the new coordinates \((x', y')\) will be:

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

This means that the point \((x, y)\) after a 270-degree clockwise rotation about the origin becomes \((-y, x)\).