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?

Answer 1

To rotate a point \((x, y)\) counterclockwise about the origin by an angle \(\theta\), you can use the following transformation formulas:

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

For a rotation of 270 degrees, we first convert the angle to radians. The angle of 270 degrees is equivalent to \(\frac{3\pi}{2}\) radians.

We know that:
- \(\cos\left(\frac{3\pi}{2}\right) = 0\)
- \(\sin\left(\frac{3\pi}{2}\right) = -1\)

Now substituting these values into the rotation formulas:

1. For the new x-coordinate \(x'\):
\[
x' = x \cos\left(\frac{3\pi}{2}\right) - y \sin\left(\frac{3\pi}{2}\right) = x \cdot 0 - y \cdot (-1) = y
\]

2. For the new y-coordinate \(y'\):
\[
y' = x \sin\left(\frac{3\pi}{2}\right) + y \cos\left(\frac{3\pi}{2}\right) = x \cdot (-1) + y \cdot 0 = -x
\]

Now, substituting in the point \((-4, 5)\):
- \(x = -4\)
- \(y = 5\)

Calculating \(x'\) and \(y'\):
- \(x' = y = 5\)
- \(y' = -x = -(-4) = 4\)

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

\[
\boxed{(5, 4)}
\]