Question

A quadrilateral is drawn on a coordinate plane with vertices G(10,−3) ,  H(2,6) , J(−4,0) , and K(0,−5) . Use rotation mapping to determine the location of  J after a −270 -degree (clockwise) rotation about the origin.(1 point)

Answer 1

[AI-generated answer — Bot GPT 3.5]

To perform a −270-degree clockwise rotation about the origin, we can use the following mapping rule:

(x', y') = (x*cosθ - y*sinθ, x*sinθ + y*cosθ)

For a −270-degree rotation, θ = −270 degrees.

Now, let's apply the mapping rule to point J(-4, 0):

(x', y') = (-4*cos(-270) - 0*sin(-270), -4*sin(-270) + 0*cos(-270))

To simplify the calculation, we need to convert −270 degrees to radians:

θ(rad) = θ(degrees) * π/180
θ(rad) = -270 * π/180
θ(rad) = -3π/2

(x', y') = (-4*cos(-3π/2) - 0*sin(-3π/2), -4*sin(-3π/2) + 0*cos(-3π/2))

Using the trigonometric identities cos(-θ) = cos(θ) and sin(-θ) = -sin(θ), we get:

(x', y') = (-4*cos(3π/2), -4*(-sin(3π/2)))
(x', y') = (0, 4)

Therefore, after a −270-degree rotation about the origin, the location of J will be (0, 4).

Answer 2

[AI-generated answer — Step-by-Step Bot]

To determine the location of J after a -270-degree (clockwise) rotation about the origin, we can follow these steps:

1. Draw a coordinate plane.
2. Plot the coordinates of the original points: G(10, -3), H(2, 6), J(-4, 0), and K(0, -5).
3. Locate the origin, which is (0, 0).
4. Draw a line from the origin to J(-4, 0).
5. Measure the angle between the line and the positive x-axis (clockwise). In this case, the angle is 180 degrees.
6. Subtract the given rotation angle of -270 degrees from the original angle. -270 degrees + 180 degrees = -90 degrees.
7. Rotate the line counterclockwise by 90 degrees, which is equivalent to rotating it clockwise by -90 degrees.
8. Measure the new angle between the rotated line and the positive x-axis. This new angle is -90 degrees + 180 degrees = 90 degrees.
9. Draw a line with the same length as the original line, following the new angle of 90 degrees.
10. Locate the endpoint of the new line, which represents the new position of J after the rotation.
11. The new coordinates of J after the -270-degree rotation about the origin are (0, -4).

Therefore, J(−4, 0) moves to J'(0, -4) after a -270-degree (clockwise) rotation about the origin.

Answer 3

[AI-generated answer — Explain Bot]

To find the location of J after a -270 degree (clockwise) rotation about the origin, we can use the concept of rotation mapping.

First, plot the given points G(10, -3), H(2, 6), J(-4, 0), and K(0, -5) on a coordinate plane.

Next, we need to understand the rotation mapping. For a rotation of -270 degrees about the origin, each point (x, y) is transformed to a new point (-y, x).

Applying this mapping to point J(-4, 0), we get:
New x-coordinate = -y = -0 = 0,
New y-coordinate = x = -4.

Therefore, after a -270-degree rotation about the origin, the new location of J is (0, -4).