Explain why transformations result in images that are congruent to the original shapes
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Transformations are geometric operations that change the position, orientation, and/or size of a shape. There are four main types of transformations: translations, rotations, reflections, and dilations. When a shape undergoes any of these transformations, the resulting image is congruent to the original shape.
Congruence means that two shapes have the same shape and size, regardless of their position or orientation in space. In other words, if two shapes are congruent, they can be superimposed on top of each other perfectly, without any gaps or overlaps.
To explain why transformations result in congruent images, let's look at each type of transformation:
1. Translations: A translation moves a shape in a specific direction without changing its size or shape. To perform a translation, you shift the shape vertically and/or horizontally. Since the shape isn't being distorted or changed in any way, the resulting image is congruent to the original shape.
2. Rotations: A rotation turns a shape around a fixed point, called the center of rotation. When you rotate a shape, each point on the shape moves along a circular path, while maintaining the same distance from the center of rotation. Therefore, the shape's size and shape remain unchanged, resulting in a congruent image.
3. Reflections: A reflection flips a shape over a line called the line of reflection. The line of reflection acts as a mirror, causing each point on the shape to be reflected across the line. Since the shape is mirrored or flipped, its size and shape remain the same, making the resulting image congruent to the original shape.
4. Dilations: A dilation changes the size of a shape but keeps its shape and proportions intact. It involves stretching or shrinking the shape either towards or away from a fixed point called the center of dilation. The ratio of the original shape's dimensions to the dimensions of the dilated shape remains constant. As a result, while the size may change, the shape remains the same, leading to congruent images.
In summary, transformations result in congruent images because they either preserve the original shape and size (translations and reflections) or only change the size while maintaining shape and proportions (rotations and dilations). By understanding the nature of each transformation, we can see why the resulting images are congruent to the original shapes.