Question

What is the new point when you rotate a geometric figure with point (5, 8) 270 degrees counterclockwise about the origin? (1 point)
0 (8, - 5)
0 (8,5)
〇 （-8， -5）
0 (-5,8)

Answer 1

To rotate a point \((x, y)\) by an angle \(\theta\) counterclockwise about the origin, you can use the following transformation:
\[
(x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)
\]

For a rotation of 270 degrees counterclockwise, we have:
\[
\theta = 270^\circ
\]
The cosine and sine of 270 degrees are:
\[
\cos(270^\circ) = 0 \quad \text{and} \quad \sin(270^\circ) = -1
\]

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

Thus, the new point after the rotation is:
\((8, -5)\)

So, the correct answer is:
\((8, -5)\)