Math - Transformations / What they're?

 Here I continue with the talk about transformations, and now we're going to learn about the different types of transformations.

• Translation
• Rotation
• Reflection
• Dilation

 - Translation:

Let's understand the Translation, translation is the most simple of the four, you just need to sum the translation factor, and boom, you have made a translation, here's an example:

    "You have a point A, in the position (5, 10), translate this point into the position (10, 5)"

To perform this translation you get the difference in the x-axis and y-axis

On the x-axis, it's added 5 units, and on the y-axis, it's removed 5 units.

So to translate point A(5, 10) to point A'(10, 5) you just need to make this: A(5+5, 10-5).

One more thing that you need to remember is: left and right = x-axis, up and down = y-axis

Left and Down = negative

Right and Up = positive

Usually, the representation of a translation is: translate A(5, 10) into A'(x+5, y-5), where you make the calculation based on A x-axis and A y-axis.

If you want to translate a geometric figure like a triangle, you just need to translate every point.

For example: translate the triangle A(-3, -4) B(-5, 5) C(3, -3) by (x+3, y-1) or 3 units to the right and 1 unit down.

A(-3, -4) -> A'(0, -5)

B(-5, 5) -> B'(-2, 4)

C(3, -3) -> C'(6, -4)

 - Rotation:

Rotation is also easy, when doing a "rotation" it means that you need to turn the figure around the center. When rotating a figure, some things are the same, for example, the distance between a point in the figure to the center, as you can draw a circle where the distance from the center to the point is the radius.

To know if a rotation is clockwise or counter-clockwise you just need to know if the angle is negative or positive, negative for clockwise and negative for counter-clockwise.

Another thing that is good to know is that in rotations centered at the origin:

Rotation of 90º:
(x,y) becomes (-y,x)

Rotation of 180º:
(x,y) becomes (-x,-y)

Rotation of 270º:
(x,y) becomes (y,-x)

 - Reflection:

Reflection is one of the easiest transformations, you basically just need to do a 'flip' in the image, for example, if the line of reflection is on the x-axis and you have a point (3, 2) when doing the reflection, this point becomes: (3, -2) if the line is on the y-axis the point(3, 2) becomes (-3, 2), all the points need to have the same distance from the line but on the other.

 - Dilation:

Dilation, in a technic way, means: changing the size of an object without changing its shape.

To dilate a figure you need to multiply each side by a scale value, just like maps.

For example, to dilate a triangle 3, 4, 5 by 2 you need to multiply 3 * 2, 4 * 2, and 5 * 2.

If the figure is in a Cartesian Plane, just like the rotation, you need to have an origin, so you don't multiply the sides, but the distance to the origin by the scale.

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If you want to learn more about transformations, I recommend you this link: https://www.khanacademy.org/math/geometry-home/transformations