Math Problem Statement

What is the image of the point left parenthesis, minus, 4, comma, 8, right parenthesis(−4,8) after a rotation of 90, degrees90∘ counterclockwise about the origin?

Solution

To find the image of the point (4,8)(-4, 8) after a 90° counterclockwise rotation about the origin, we use the rotation formula:

Rotation Formula

For a 90° counterclockwise rotation about the origin, a point (x,y)(x, y) is transformed to (y,x)(-y, x).

Applying the Formula

Given the point (4,8)(-4, 8):

  • x=4x = -4
  • y=8y = 8

The new coordinates after a 90° counterclockwise rotation will be: (x,y)=(y,x)(x', y') = (-y, x)

Substitute y=8y = 8 and x=4x = -4: (x,y)=(8,4)(x', y') = (-8, -4)

Conclusion

The image of the point (4,8)(-4, 8) after a 90° counterclockwise rotation about the origin is (8,4)(-8, -4).

Would you like more details on this or have any questions?

Here are 5 related questions to further explore this concept:

  1. What is the image of the point (5,3)(5, -3) after a 90° counterclockwise rotation about the origin?
  2. How would you describe a 270° clockwise rotation's effect on a point's coordinates?
  3. What is the result of rotating a point (2,7)(2, -7) by 180° about the origin?
  4. How does the distance from the origin change after rotating a point 90° counterclockwise?
  5. Can you find the image of a point (a,b)(a, b) after a 360° rotation about the origin?

Tip: Remember that a 90° counterclockwise rotation changes the sign of the y-coordinate and swaps the coordinates.

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Math Problem Analysis

Mathematical Concepts

Geometry
Coordinate Geometry
Rotations

Formulas

Rotation formula for 90° counterclockwise rotation about the origin

Theorems

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Suitable Grade Level

Grades 9-12