Question

Given a point on a line segment (2,3), what is the new point when the point is rotated counterclockwise about the origin 270 degrees?
*
1 point
(3,-2)
(2,-3)
(-2,3)
(-3,2)

Answer 1

To rotate a point \((x, y)\) counterclockwise about the origin by \(\theta\) degrees, the following formulas are used:

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

For a rotation of 270 degrees, we first convert this to radians, which equals \(-90\) degrees (as rotating 270 degrees counterclockwise is the same as rotating 90 degrees clockwise).

Using the values:
- \(\cos(-90^\circ) = 0\)
- \(\sin(-90^\circ) = -1\)

Now, substituting the original point \((2, 3)\):

\[
x' = 2 \cdot 0 - 3 \cdot (-1) = 0 + 3 = 3
\]
\[
y' = 2 \cdot (-1) + 3 \cdot 0 = -2 + 0 = -2
\]

So the new point after rotating \((2, 3)\) counterclockwise by 270 degrees is \((3, -2)\).

Thus, the answer is:

**(3, -2)**