The point Z(–6,6) is rotated 270° counterclockwise around the origin. What are the coordinates of the resulting point, Z
I t is originally 90+45 = 135 deg cc from the x axis
spin 270 more ----> 405 total
405 - 360 = 45
so 45 degrees above x axis in quadrant 1
(6, 6)
The point C(4,5) is rotated 270° counterclockwise around the origin. What are the coordinates of the resulting point, C'?
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The point J(
–
2,2) is rotated 270° clockwise around the origin. What are the coordinates of the resulting point, J'?
When a point is rotated 270° clockwise, it is the same as rotating 90° counterclockwise.
To rotate a point 90° counterclockwise around the origin, you first swap the x and y coordinates and then negate the new x coordinate.
So, for point J(-2,2), we swap the coordinates to get (2,-2) and then negate the x-coordinate to get (-2,-2).
Therefore, the coordinates of the resulting point J' are (-2,-2).
Well, Z, let's see if we can rotate your life a little bit. When you're rotated 270° counterclockwise around the origin, you end up doing some serious spinning. So, hold on tight!
Now, to find your new coordinates, we need to think about the process of rotating. Since you're starting at -6 on the x-axis and 6 on the y-axis, let's break it down. Rotating 270° counterclockwise means you're basically doing a three-quarter turn.
So, let's get dizzy together! Starting with your x-coordinate of -6, we'll rotate it 270° counterclockwise. A full turn is 360°, so subtracting 270° from that gives us 90°. Now, in terms of degrees, 90° means you would end up on the y-axis. So your new x-coordinate is 0.
Next, let's tackle your y-coordinate of 6. Again, we'll rotate it 270° counterclockwise, which leaves us with 90°. Now, in terms of degrees, 90° means you would end up on the negative x-axis. So your new y-coordinate is -6.
Putting it all together, the coordinates of the resulting point Z are (0, -6). You went for a wild spin, but it looks like you ended up somewhere interesting!
To find the coordinates of the point Z after it is rotated 270° counterclockwise around the origin, we can use the rotation formula.
The rotation formula for a point (x, y) rotated counterclockwise by an angle θ around the origin is:
(x', y') = (x * cos(θ) - y * sin(θ), x * sin(θ) + y * cos(θ))
In this case, the point Z has coordinates (-6, 6) and we want to rotate it by 270° counterclockwise.
Plugging the values into the formula, we get:
(x', y') = (-6 * cos(270°) - 6 * sin(270°), -6 * sin(270°) + 6 * cos(270°))
To further simplify the calculation, we can use some trigonometric identities. Since cos(270°) = 0 and sin(270°) = -1, the formula becomes:
(x', y') = (-6 * 0 - 6 * (-1), -6 * (-1) + 6 * 0)
Calculating further:
(x', y') = (0 + 6, 6 - 0)
Therefore, the coordinates of the resulting point Z after rotating 270° counterclockwise around the origin are (6, 6).