Isometries, in the context of geometry, refer to a type of transformation that preserves distances between points. In other words, when a geometric shape undergoes an isometry, its size and shape remain unchanged. However, it’s orientation, or position in space will change. Because both the size and shape of the object are unchanged, these isometric transformations are sometimes called rigid transformations. There are several types of isometries, including translations, rotations, and reflections. All are explained in this downloadable PDF. If visualizing how figures moves around and change shape is still tough, it helps to have tracing paper handy so you can compare images to see what happened. If it's just not clicking, and you need help, give me a shout.

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To understand isometries you need to know the difference between a pre-image and image. The pre-image is the original figure before undergoing any transformation. It is the geometric shape in its unaltered state. In contrast, the image is the resultant figure after the transformation has occurred. Understanding these terms gives a clear before-and-after perspective that helps you to see which transformations have occurred.

Another idea to get under your belt is image mapping. Image mapping refers to the process of applying the same mathematical operation to every point on a figure. Mappings alter the positions of points on geometric objects. Each point in the original object is mapped to a new location based on the transformation applied.

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Translations involve moving a shape without changing its orientation or size. Every point on the shape moves the same distance and in the same direction. Another way to say this is that a translation is a coordinated spatial shift of all points on a figure. For instance, imagine a point (5,-8)(x,y) on a Cartesian plane. If you move this point up 3 units and over two units, (3,2)(a,b), the image mapping would take the original point to the new point (5+3),(-8+2) or (8,6)(x+a,y+b). The amount and direction that the point or figure moves is determined by a rule on a Cartesian coordinate system which denotes the precise shifts along the x and y axes that each point experiences.

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Rotation transformations involve turning a shape around a fixed point known as the center of rotation, and a rotation about a point is an isometry. The shape maintains its size and shape, but is reoriented. The center of rotation may or may not be on the figure. The number of degrees a figure rotates clockwise or counter clockwise is the angle of rotation.

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A reflection is a transformation formed by flipping a shape over a specified line known as the line of reflection, creating a mirror image. If the vertices of the pre-image read clockwise, those of the image read counter-clockwise and vice versa. Additionally, there is a point on the line of reflection that both the pre-image image and the image share. The line of reflection is the perpendicular bisector of lines between corresponding points on the image and pre-image.

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Dilations are not rigid and involve uniformly stretching or shrinking the size of a figure while preserving its overall shape. When a pre-image undergoes a dilation, the image can be either larger or smaller than the original. If it is larger, the scale factor (n) is greater than 1 and the transformation is called an enlargement or expansion. If it is smaller, then n < 1 and the transformation is called a reduction or compression. Determining the scale factor involves comparing corresponding lengths on the original and transformed shapes. This comparison gives the degree of enlargement or reduction. Another way to think about this is that the scale factor is the ratio of the length of a side in the image to the length of the corresponding side in the preimage. All points in the figure will either move toward or way from the center of dilation when the transformation occurs.

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