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 counterclockwise about the origin, we can use the rotation formula. For a given point \((x, y)\) and an angle of \(\theta\), the new coordinates \((x', y')\) after rotation can be calculated using the following equations:

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

In this case, the angle \(\theta\) is 270 degrees.

First, we need to find the cosine and sine of 270 degrees:

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

Now, we can substitute the 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 coordinates after rotating the point \((-4, 5)\) counterclockwise about the origin by 270 degrees are \((5, 4)\).