Question

Point M (0,4) is rotated about the origin 270 degrees. What are the coordinates of M′

Answer 1

To find the new coordinates of point M (0, 4) after it is rotated about the origin by 270 degrees, we can use the rotation transformation formulas.

A rotation of 270 degrees is equivalent to a rotation of -90 degrees. The formula for rotating a point \((x, y)\) 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 -90 degrees (or 270 degrees), \(\theta = -90^\circ\) which corresponds to:

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

Plugging in the values for point M (0, 4):

\[
x' = 0 \cdot 0 - 4 \cdot (-1) = 0 + 4 = 4
\]

\[
y' = 0 \cdot (-1) + 4 \cdot 0 = 0 + 0 = 0
\]

Thus, the coordinates of \(M'\) after the rotation are:

\[
M' (4, 0)
\]

So, the new coordinates after rotating point M (0, 4) by 270 degrees about the origin are \(\boxed{(4, 0)}\).