Glory to God in the highest; and on earth, peace to people on whom His favor rests!

- Luke 2:14

The Joy of a Teacher is the Success of his Students.

- Samuel Chukwuemeka

# Welcome to Geometry Transformations Calculators

I greet you this day,

You may use these calculators to check your answers. You are encouraged to solve the questions graphically (by construction), and check your answers algebraically (by formulas). These topics are covered in my Videos on Geometry Transformations.

I wrote the codes for these calculators using Javascript, a client-side scripting language. Please use the latest Internet browsers. The calculators should work.
As at the moment, only the calculators for the __Translations, Reflections, and Rotations are completed.__ Please check back for the remaining ones later.

Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. Should you need to contact me, please use the form at the bottom of the page. Thank you for visiting!!!

**Samuel Chukwuemeka** (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

- Translation (Sliding or Gliding)
- For any point, (x
_{1}, y_{1}); - A translation vector of <x, y> is the same as the translation rule of (x
_{1}+ x, y_{1}+y) - Say: preimage = (x
_{1}, y_{1}), translation vector = <x, y>, and image = (x_{2}, y_{2}); - then: x
_{2}= x_{1}+ x and y_{2}= y_{1}+ y

__Given__: preimage, translation vector

__To Find__: image

__Given__: image, translation vector

__To Find__: preimage

__Given__: preimage, image

__To Find__: translation vector

- 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), and an angle of rotation, θ
- the image would be (x
^{'}, y^{'}) where x^{'}= x cosθ - y sinθ and 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 (x
_{1}, y_{1}), the center of rotation as any point (x, y), and an angle of rotation, θ - the image would be (x
_{1}^{'}, y_{1}^{'}) where - x
_{1}^{'}= (cosθ(x_{1}- x) - sinθ(y_{1}- y)) + x - y
_{1}^{'}= (cosθ(y_{1}- y) + sinθ(x_{1}- x)) + y