Question

Use the image to answer the question.

An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 8 to 8 in increments of 1. Four triangles are plotted on the graph. The four triangles plotted are triangle A B C, triangle A prime B prime C prime, triangle A double prime B double prime C double prime, and triangle A triple prime B triple prime C triple prime. The triangle A B C has its vertices marked with closed points at A left parenthesis 2 comma 4 right parenthesis, B left parenthesis 6 comma 6 right parenthesis, and C left parenthesis 6 comma 2 right parenthesis. The triangle A prime B prime C prime has its vertices marked with closed points at A prime left parenthesis negative 4 comma 2 right parenthesis, B prime left parenthesis negative 6 comma 6 right parenthesis, and C prime left parenthesis negative 2 comma 6 right parenthesis. The triangle A double prime B double prime C double prime has its vertices marked with closed points at A double prime left parenthesis negative 2 comma negative 4 right parenthesis, B double prime left parenthesis negative 6 comma negative 6 right parenthesis, and C double prime left parenthesis negative 6 comma negative 2 right parenthesis. The triangle A triple prime B triple prime C triple prime has its vertices marked with closed points at A triple prime left parenthesis 4 comma negative 2 right parenthesis, B triple prime left parenthesis 6 comma negative 6 right parenthesis, and C triple prime left parenthesis 2 comma negative 6 right parenthesis.

Given point A (2, 4)
on the triangle Quadrant I, where is the new point if the triangle is rotated 270 degrees counterclockwise?

(1 point)
Responses

(−4, 2)
left parenthesis negative 4 comma 2 right parenthesis

(4,−2)
left parenthesis 4 comma negative 2 right parenthesis

(−4,−2)
left parenthesis negative 4 comma negative 2 right parenthesis

(−2,−4)
left parenthesis negative 2 comma negative 4 right parenthesis
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Answer 1

To determine the new coordinates of point A (2, 4) after a 270-degree counterclockwise rotation, we can use the rotation transformation formulas for the coordinate plane.

The general formula for rotating a point (x, y) counterclockwise by an angle θ is given by:
- x' = x * cos(θ) - y * sin(θ)
- y' = x * sin(θ) + y * cos(θ)

For a 270-degree rotation counterclockwise, we can convert this to radians as follows:
270 degrees = \( \frac{3\pi}{2} \) radians

Using this angle, the cosine and sine values are:
- cos(270°) = 0
- sin(270°) = -1

Now we can apply the formulas to point A (2, 4):
- x' = 2 * 0 - 4 * (-1) = 0 + 4 = 4
- y' = 2 * (-1) + 4 * 0 = -2 + 0 = -2

Thus, after a 270-degree counterclockwise rotation, point A (2, 4) moves to the new point:
(4, -2)

So the correct response is:
(4, −2)