Question

M(0,4) after being rotated 270 degrees counterclockwise would make M' what?

Answer 1

To find the coordinates of point \( M' \) after rotating point \( M(0, 4) \) by 270 degrees counterclockwise around the origin, you can use the rotation formula for counterclockwise rotation by an angle \( \theta \):

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

For a rotation of 270 degrees, we can convert this to radians:

\[
270^\circ = \frac{3\pi}{2} \text{ radians}
\]

Now, the cosine and sine values for 270 degrees are:

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

Substituting the original coordinates \( (x, y) = (0, 4) \) into the formula, we calculate \( (x', y') \):

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

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

\[
M'(4, 0)
\]

So, the final answer is \( M' = (4, 0) \).