Question

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

Answer 1

To rotate the point Q(−5,−9) by −270° (which is equivalent to a positive 90° rotation) about the origin, we can use the rotation formulas.

The formula for rotating a point \((x, y)\) by \( \theta \) degrees counterclockwise is given by:
\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]

For a rotation of 90°, the cosine and sine values are:
\[
\cos(90°) = 0 \quad \text{and} \quad \sin(90°) = 1
\]

Now, substituting \( x = -5 \) and \( y = -9 \) into our formulas:

1. For \(x'\):
\[
x' = -5 \cdot 0 - (-9) \cdot 1 = 0 + 9 = 9
\]

2. For \(y'\):
\[
y' = -5 \cdot 1 + (-9) \cdot 0 = -5 + 0 = -5
\]

Thus, the coordinates of the rotated point \( Q' \) are \( (9, -5) \).

Therefore, the location of \( Q' \) after the rotation is \( (9, -5) \).