Eli can’t see why a size transformation preserves orientation, but a similitude does not. How could you help him understand this concept?
To help Eli understand why a size transformation preserves orientation while a similitude does not, I would explain the concepts of size transformation and similitude, and then provide examples to illustrate the differences.
1. Size Transformation:
A size transformation, also known as a scaling or dilation, is a geometric transformation that changes the size of an object while preserving its shape. It involves multiplying the coordinates of each point in the object by a scale factor. The scale factor determines the amount and direction of the size change.
2. Similitude:
A similitude is a geometric transformation that combines a size transformation and a rigid motion (translation, rotation, or reflection). It involves changing the size of an object and moving it in space, while preserving its shape. The scale factor and the rigid motion together determine the similitude.
Now, let's consider why a size transformation preserves orientation but a similitude does not:
Orientation refers to the arrangement of points in an object or a figure. It includes the concept of clockwise and counterclockwise direction.
In a size transformation, only the size changes while the shape remains the same. Since the shape is preserved, the arrangement of points also remains unchanged, including their orientation. For example, if you scale up or down a square, the angles between its sides will remain the same, and the orientation will be preserved.
On the other hand, a similitude involves both a size transformation and a rigid motion. The rigid motion can include rotations or reflections, which can change the arrangement of points in the object. Depending on the specific rigid transformation applied, the orientation of the object may change. For example, if you rotate a square, the angles between its sides will change, and the orientation will not be preserved.
Therefore, while a size transformation only affects the size of an object, preserving its orientation, a similitude combines size transformation with rigid motion, leading to potential changes in orientation.