Question

The point Q(−5,−9)
is rotated −270°
about the origin. Select the location of Q′
. (1 point)
Responses

Q′(9,−5)
upper Q prime left parenthesis 9 comma negative 5 right parenthesis

Q′(−5,9)
upper Q prime left parenthesis negative 5 comma 9 right parenthesis

Q′(5,9)
upper Q prime left parenthesis 5 comma 9 right parenthesis

Q′(9,5)

Answer 1

To find the new coordinates of the point Q(−5,−9) after rotating it −270° about the origin, we can use the rotation transformation formulas:

For a rotation of θ degrees about the origin, the coordinates (x, y) transform as follows:

- New x-coordinate: \( x' = x \cdot \cos(\theta) - y \cdot \sin(\theta) \)
- New y-coordinate: \( y' = x \cdot \sin(\theta) + y \cdot \cos(\theta) \)

First, let's convert −270° to an equivalent positive angle. Since rotating −270° is the same as rotating +90° (because rotating in the negative direction is the same as going back to zero and then forward 90°), we will use +90° for our calculations.

For 90°:
- \( \cos(90°) = 0 \)
- \( \sin(90°) = 1 \)

So the transformation simplifies to:
- \( x' = 0 \cdot x - 1 \cdot y = -y \)
- \( y' = 1 \cdot x + 0 \cdot y = x \)

Now applying this to Q(−5,−9):
- \( x' = -(-9) = 9 \)
- \( y' = -5 \)

Thus, the new coordinates after a rotation of −270° (or +90°) are:
- \( Q' = (9, -5) \)

So the correct answer is:
**Q′(9,−5)**.