Question

what is the new point when you rotate a geometric figure with point (5, 8) 270 degrees counterclockwise about the orgin

Answer 1

To rotate a point \( (x, y) \) counterclockwise around the origin by \( \theta \) degrees, you can use the rotation formulas:

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

For a 270-degree counterclockwise rotation, we note that:

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

Now, substituting in the values for the point \( (5, 8) \):

\[
x' = 5 \cdot 0 - 8 \cdot (-1) = 0 + 8 = 8
\]
\[
y' = 5 \cdot (-1) + 8 \cdot 0 = -5 + 0 = -5
\]

Thus, the new coordinates after rotating the point \( (5, 8) \) 270 degrees counterclockwise about the origin are:

\[
(8, -5)
\]