Question

Given a point on a line segment (−4, 5)
, what is the new point when the point is rotated counterclockwise about the origin 270 degrees?(1 point)
(
)

Answer 1

To determine the new coordinates of the point \((-4, 5)\) after rotating it counterclockwise about the origin by 270 degrees, we can use the rotation formula.

When rotating a point \((x, y)\) counterclockwise by an angle \(\theta\), the new coordinates \((x', y')\) are given by:

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

For a rotation of 270 degrees, we first need the values of \(\cos(270^\circ)\) and \(\sin(270^\circ)\):
- \(\cos(270^\circ) = 0\)
- \(\sin(270^\circ) = -1\)

Plugging these values into the rotation formulas, we get:

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

Now, substituting in the coordinates of our point \((-4, 5)\):

1. \(x' = 5\) (which is \(y\))
2. \(y' = -(-4) = 4\) (which is \(-x\))

Thus, after rotating the point \((-4, 5)\) counterclockwise about the origin by 270 degrees, the new point is:

\[
(5, 4)
\]