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)
(
,
)
Remaining Attempts : 3

Answer 1

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

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

For a 270-degree rotation (which is equivalent to a -90-degree rotation), we can calculate the sine and cosine values:

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

Now we can substitute the initial point \((-4, 5)\) into the rotation formulas:

\[
x' = -4 \cdot 0 - 5 \cdot (-1) = 0 + 5 = 5
\]
\[
y' = -4 \cdot (-1) + 5 \cdot 0 = 4 + 0 = 4
\]

Thus, the new point after a 270-degree counterclockwise rotation about the origin is \((5, 4)\).

Therefore, the answer is:

\[
(5, 4)
\]