Given a point (9,-3) on a geometric figure, what is the new point when the figure is rotated counterclockwise about the origin 180 degrees?
To rotate a point counterclockwise about the origin, we can use the formula:
\[ (\text{x'}, \text{y'}) = (\text{x}\cos(\theta) - \text{y}\sin(\theta), \text{x}\sin(\theta) + \text{y}\cos(\theta)) \]
where \((\text{x}, \text{y})\) is the original point, \((\text{x'}, \text{y'})\) is the new point after rotation, and \(\theta\) is the angle of rotation.
In this case, the original point is \((9, -3)\) and we want to rotate it 180 degrees.
Inserting this information into the formula, we have:
\[\begin{aligned} (\text{x'}, \text{y'}) &= (9\cos(180^\circ) - (-3)\sin(180^\circ), 9\sin(180^\circ) + (-3)\cos(180^\circ)) \\ &= (9 \cdot (-1) - (-3) \cdot 0, 9 \cdot 0 + (-3) \cdot (-1)) \\ &= (-9, 3) \end{aligned}\]
Therefore, the new point after rotating counterclockwise about the origin 180 degrees is \((-9, 3)\).
To rotate a point counterclockwise about the origin, you can use the following formula:
x' = x * cos(angle) - y * sin(angle)
y' = x * sin(angle) + y * cos(angle)
In this case, the point is (9, -3) and the rotation angle is 180 degrees.
First, let's find the new x-coordinate:
x' = 9 * cos(180) - (-3) * sin(180)
x' = 9 * (-1) - (-3) * 0
x' = -9
Next, let's find the new y-coordinate:
y' = 9 * sin(180) + (-3) * cos(180)
y' = 9 * 0 + (-3) * (-1)
y' = -3
Therefore, the new point after rotating (9,-3) counterclockwise about the origin by 180 degrees is (-9, -3).