Transformations (GEOMETRY)

Transformation means something’s changing, transforming.

Refer to Khan academy: Transformations

Types of transformations:

• Translations: Slide or move the shape. In geometry, a translation moves a thing up and down or left and right.
• Rotation: Turn or rotate the shape. In geometry, rotations make things turn in a cycle around a definite center point.
• Reflection: Flip or mirror the shape. A reflection is a transformation that acts like a mirror: It swaps all pairs of points that are on exactly opposite sides of the line of reflection.
• Dilation: Expand or shrink the shape.

And,
you can group above types of transformations into two groups: Rigid transformations and Dilations

Rigid transformations and Congruent

Rigitd transformations means you play around the shape without expanding or shrinking it.
Or say, without dilation, all translation/rotation/reflection would be rigid transformation.

Congruent is the shape after you "rigid transformed`.
Or say, without dilation, after all translation/rotation/reflection, the shape is called "congruent" to the original shape.

Khan lecture.

Dilations

Dilation, just a fancy word for "resizing", or "scaling".

Notice that: Dilation DOES NOT change any angle within the shape.
Don’t get confused with a horizontal stretch, which does change both sides and angles.

Similarity

When you resize, or say dilate a shape, you call the new shape similar to the original one.
If nothing changed with the shape, you call it congruent to the original one.

Scale factor

Means how much you scale the shape, like 2 times bigger, or 2/3 smaller.

Notice:

• The scale factor is of the length of the shape, NOT the area of it.

Scale factors and area

When you scale the shape, the area of the new shape is (scale facto)² times to the original one.

Khan lecture: Scale factors and area

For Shape A and scaled shape A’, it leads to two practical conclusions:

• If we know the scale factor is x, then the area of A is x² times to the original one.
• If we know the ratio (area of A') ÷ (area of A) is x, then the scale factor is √x

Example

Solve:

• New area G is 1/9 of F
• So the (scale factor)² = 1/9, which results the scale factor = 1/3

Dilation Center

“Dilate the shape ABOUT a point P”, means take the point as a center to dilate the shape.

How does it work? As the picture below, just simply scale the distance from each vertex(point) of the shape to the point.

How to find the dilation center?

The point P and it’s image and dilation center, should be IN ONE LINE !

In the example below, you should forget about the origin but set the P point as origin and count the distance of each vertex of the triangle: