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