Geometry for Elementary School/Transformation

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Geometry for Elementary School
Symmetry Transformation Coordinates

Transformation is when we change the size, orientation, and/or position of a shape. Note that transformation is usually done on graph paper to avoid excessive meaurements and ensure accuracy.

Reflection[edit]

Reflectional transformation.svg

Reflection is when a shape is reflected along an axis to produce a reflectionally symmetrical figure. The axis of reflection is also the axis of symmetry of the new figure.

Look at the diagram on the right. Imagine you are given the left part. How can you reflect the figure? First, Find out the distance between A and the axis. That's four. Then find point A′ (pronounced 'A prime'), which should be the same distance from the axis but on the different side. Look at the figure that is on the right of the figure. If you found that point, you're right. Put a little cross there - if your teacher doesn't let you do it, just erase it later! Now do the same for the other two points. Join them up. Now you have formed the reflected figure!

A common mistake while reflecting a figure is forgetting to mark the points. That costs you a lot in marks, so don't make this mistake! Also, never mark the point wrongly. Remember to add the ' ′ ' symbol and check if the points are corresponding!

Rotation[edit]

Rotation is the most difficult kind of transformation. It requires rotating a figure with reference to a single point. Therefore, you will only be asked to rotate 90°, 180° and 270° at this stage. (360° is meaningless, but if you do manage to come across it you're very lucky!) There are three things that we need to note before rotating a figure.

  • The number of degrees to rotate
  • Whether you should rotate something clockwise or anticlockwise
  • Where the centre of rotation is
Rotational transformation.svg

Let us look at the example on the left. Imagine we are only given the top, left one. We need to rotate it in a number of ways.

  1. 90° anticlockwise through the point in the middle
  2. 180° anticlockwise through the point in the middle
  3. 270° anticlockwise through the point in the middle
  4. 90° clockwise through the point in the middle
  5. 180° clockwise through the point in the middle
  6. 270° clockwise through the point in the middle

Look at #1 and #6. If you look at them carefully, you should be able to see that #1 and #6 are actually the same! This can be explained through the example. To translate triangle ABC 90° anticlockwise through the point in the middle, we can use point A first. Point a is three squares to the left and one square to above the centre of rotation, so we can rotate that 90 degrees. This cannot be explained in words, so try looking at the figure on the right. As you can see, A and A′ are the same distance from the centre of rotation. We do the same to the other two points, producing a triangle that looks like the one in the figure. If we do the same for number 6, you will produce an identical triangle! The same goes for points number 3 and 4.

What about 2 and 5? As we can see, they both involve translating 180°; however, one is clockwise and the other is anticlockwise. Let's try rotating it clockwise first. You should get the triangle A′′B′′C′′ in the figure. Then do the same for the anticlockwise. Do they look the same?

Now that we have tried rotating 6 possibilities from one centre of rotation, and figuring out there is only three, can you try rotating triangle ABC through point A? If you are reading this as an e-book, please copy out triangle ABC on a piece of graph paper. If you are using the print version, you can draw the new triangle directly on your book. Name your new triangle DEF.

Translation[edit]

Translation is a simple transformation. Put simple, translation is the change of the position of a shape. For example, if we want to transform a figure five units to the left, then we just move it five units to the left.

The process of translating is very easy. Imagine we have a right-angled triangle ABC on a piece of graph paper. Our task is to translate it four squares upwards and two squares to the left. We take the vertex of the right angle, named A, as our point, move it four squares upwwards, then two squares to the left. Draw a little cross there. and mark it A′. Now we re-create the shape by referring to the original shape. Remember to name the points correctly.

We are often asked to trace back the translation gone through in a given translation. We are given the original figure and the new figure. When dealing with this type of questions, it may be helpful to use a point like we did above. Take the same right-angled triangle. We see that A has been translated four squares upwards. So we write: Translate A upwards four squares. Then we see that A has been translated two squares to the left, so we write, then translate A two squares to the left. Then we have finished the question!

Enlargement and reduction[edit]

Table of conclusion[edit]

The following table shows what things are changed when a transformation is gone through.

Type of transformation Size Shape Orientation (direction) Position
Reflection Never Never Always Sometimes
Rotation Never Never Always Sometimes
Translation Never Never Never Always
Enlargement (Dilation) Always Never Never Sometimes
Reduction (Dilation) Always Never Never Sometimes

Note: For all of these changes the shape never changes. A triangle will always be a triangle, a rectangle will always be a rectangle.