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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)

(−2,−4)

(−4, 2)

(−4,−2)

1 answer

To find the new coordinates of point A (2, 4) after a 270-degree counterclockwise rotation around the origin (0, 0), we can use the rotation formula. The formula for rotating a point \((x, y)\) by \(270^\circ\) counterclockwise is given by:

\[ (x', y') = (y, -x) \]

Applying this to point A (2, 4):

  1. The x-coordinate becomes the y-coordinate: \(y = 4\).
  2. The y-coordinate becomes the negative of the x-coordinate: \(-x = -2\).

So, after the rotation:

\[ A' = (4, -2) \]

Thus, the new point after rotating point A (2, 4) by 270 degrees counterclockwise is \((4, -2)\).

The correct response is:

(4,−2)