Question

The point Q (-5, -9) is rotated -270 degree about the origin. Select the location of Q'. A: Q'(9,5) B:Q'(9,-5) C:Q' (5,9)

Answer 1

To rotate a point around the origin, we can use the rotation formula.

When rotating a point \((x, y)\) by an angle \(\theta\) in standard position, the new coordinates \((x', y')\) after rotation are given by:

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

In this case, we want to rotate the point \(Q(-5, -9)\) by \(-270\) degrees (which is equivalent to rotating \(90\) degrees counterclockwise).

First, we find the cosine and sine values for \(90\) degrees:
- \(\cos(90^\circ) = 0\)
- \(\sin(90^\circ) = 1\)

Now apply the rotation formula:

\[
x' = -5 \cdot 0 - (-9) \cdot 1 = 0 + 9 = 9
\]
\[
y' = -5 \cdot 1 + (-9) \cdot 0 = -5 + 0 = -5
\]

So the new coordinates after the rotation, \(Q'\), are \((9, -5)\).

The correct answer is **B: Q'(9, -5)**.