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A quadrilateral with vertices G(−10,1) , E(−6,−4) , O(2,0) , and M(0,4) is rotated about the origin 270 degrees (counterclockwise). Apply rotation mapping rules to find the image of M .

m'(answer here)

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To find the image of point M(0, 4) after a 270-degree counterclockwise rotation about the origin, you can use the following rotation rules:

When rotating a point (x, y) about the origin through an angle θ, the new coordinates (x', y') can be found using the formulas:

  • \( x' = x \cdot \cos(\theta) - y \cdot \sin(\theta) \)
  • \( y' = x \cdot \sin(\theta) + y \cdot \cos(\theta) \)

For a 270-degree rotation, we can use the following values:

  • \( \cos(270^\circ) = 0 \)
  • \( \sin(270^\circ) = -1 \)

Now, applying the rotation to point M(0, 4):

  • \( x = 0 \)
  • \( y = 4 \)

Now substituting into the rotation formulas:

  • \( x' = 0 \cdot 0 - 4 \cdot (-1) = 0 + 4 = 4 \)
  • \( y' = 0 \cdot (-1) + 4 \cdot 0 = 0 + 0 = 0 \)

So the coordinates after the rotation are (4, 0).

Thus, the image of M after a 270-degree rotation about the origin is:

M' = (4, 0)