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Geometry Transformations Calculators



I greet you this day,
You are encouraged to: solve the questions graphically (by construction); verify the solutions algebraically (by formulas); then use the calculators to check your answers.
These topics are covered in my Videos on Geometry Transformations.
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At the moment, only the calculators for the Translations, Reflections, and Rotations are completed. Please check back for the remaining ones later.
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Samuel Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S


  • Translation (Sliding or Gliding)
  • For any point, (x1, y1);
  • A translation vector of <x, y> is the same as the translation rule of (x1 + x, y1+y)
  • Say:
    preimage = (x1, y1),
    translation vector = <x, y>,
    image = (x2, y2);
  • then:
    x2 = x1 + x
    y2 = y1 + y

Given: preimage, translation vector

To Find: image

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Given: image, translation vector

To Find: preimage

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Given: preimage, image

To Find: translation vector

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  • Reflection (Flipping)
  • On the x-axis, y-values are zero.
    Reflecting across the x-axis: y-coordinate changes, x-coordinate remains.
  • On the y-axis, x-values are zero.
    Reflecting across the y-axis: x-coordinate changes, y-coordinate remains.
  • Reflecting across the origin: both the x-coordinate and the y-coordinate changes.
  • Reflecting across the line: x = k (k is a constant); the x-coordinate changes, y-coordinate remains.
  • Reflecting across the line: y = k (k is a constant); the y-coordinate changes, x-coordinate remains.
  • Reflecting across the line: y = x; both the x-coordinate and the y-coordinate changes.
  • For any point, (x, y);
  • Reflection across the x-axis gives (x, -y)
  • Reflection across the y-axis gives (-x, y)
  • Reflection across the origin gives (-x, -y)
  • Reflection across the line: y = x gives (y, x)
  • Reflection across the line: y = k (k is a constant) gives (x, 2k - y)
  • Reflection across the line: x = k (k is a constant) gives (2k - x, y)

Given: preimage, line of reflection

To Find: image

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reflect across the:



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Given: image, line of reflection

To Find: preimage

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reflected across the:



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Given: preimage, image

To Find: line of reflection

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  • Rotation (Turning)
  • A preimage is rotated about the "center of rotation" through an "angle of rotation".
  • If the preimage is rotated in a counterclockwise direction, the angle of rotation is positive.
  • If the preimage is rotated in a clockwise direction, the angle of rotation is negative.
  • Given:
    the preimage (x, y),
    the center of rotation as the origin (0, 0),
    an angle of rotation, θ;
    the image would be (x', y')
    where:
    x' = x cosθ - y sinθ
    y' = y cosθ + x sinθ
  • For counterclockwise rotations, use the positive value of θ
  • For clockwise rotations, use the negative value of θ
  • Given:
    the preimage (x1, y1),
    the center of rotation as any point (x, y),
    an angle of rotation, θ;
    the image would be (x1', y1')
    where:
    x1' = (cosθ(x1 - x) - sinθ(y1 - y)) + x
    y1' = (cosθ(y1 - y) + sinθ(x1 - x)) + y

Given: preimage, center of rotation (about the origin), angle of rotation

To Find: image

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Given: preimage, center of rotation (any point), angle of rotation

To Find: image

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