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Geometry Transformations Calculators

Samuel Dominic Chukwuemeka
I greet you this day,
First: read the notes.
Second: view the Videos
Third: solve the questions/solved examples.
Fourth: check your solutions with my thoroughly-explained solutions.
Fifth: check your answers with the calculators as applicable.

You are encouraged to: solve the questions graphically (by construction); verify the solutions algebraically (by formulas); then use the calculators to check your answers.
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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
  • Translated left or right:
    only the x-coordinate changes
    the y-coordinate remains

    Translated up or down:
    only the y-coordinate changes
    the x-coordinate remains
  • translated x units left:
    then:
    x2 = x1 - x
    y2 = y1
  • translated x units right:
    then:
    x2 = x1 + x
    y2 = y1
  • translated y units up:
    then:
    x2 = x1
    y2 = y1 + y
  • translated y units down:
    then:
    x2 = x1
    y2 = y1 - y

Given: preimage, translation vector

To Find: image

(, )

<, >



(, )

Given: image, translation vector

To Find: preimage

(, )

<, >



(, )

Given: preimage, image

To Find: translation vector

(, )

(, )



<,>

Given: preimage, translation rule

To Find: image

(, )

units



(, )

  • 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



(, )

Given: image, line of reflection

To Find: preimage

(, )



(, )

Given: preimage, image

To Find: line of reflection

(, )

(, )



  • 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

(, )

()

°



(, )

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

To Find: image

(, )

(, )

°



(, )

References

Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://www.samdomforpeace.com

CrackACT. (n.d.). Retrieved from http://www.crackact.com/act-downloads/

CSEC Math Tutor. (n.d). Retrieved from https://www.csecmathtutor.com/past-papers.html

DLAP Website. (n.d.). Curriculum.gov.mt. https://curriculum.gov.mt/en/Examination-Papers/Pages/list_secondary_papers.aspx

GCSE Exam Past Papers: Revision World. Retrieved April 6, 2020, from https://revisionworld.com/gcse-revision/gcse-exam-past-papers