A dilation is not a rigid transformation because it does not preserve the shape of an object. Unlike rigid transformations such as translations, reflections, and rotations, which preserve the size and shape of an object, dilation changes the size of an object while keeping its proportions intact.
One way to understand why a dilation is not a rigid transformation is by considering the concept of a scale factor. The scale factor is the ratio between the length of the original image and the length of the transformed image. When performing a dilation, each point of the original object is multiplied by the scale factor to determine its new position. This scaling effect causes the object to either expand or contract, depending on whether the scale factor is greater than or less than 1.
To illustrate this, imagine you have a square with side length 2 units. If you dilate this square by a scale factor of 2, each side of the square will be multiplied by 2, resulting in a new square with side length 4 units. However, the new square will still maintain the same shape as the original square, with all angles remaining right angles. This change in size while preserving proportions is what distinguishes dilation from rigid transformations.
Another way to understand why dilation is not a rigid transformation is by considering the concept of the center of dilation. The center of dilation is the fixed point on the plane about which the dilation occurs. When performing a dilation, the distance between each point of the original object and the center of dilation is multiplied by the scale factor to determine its new position. This means that every point of the object moves away from or towards the center of dilation, resulting in a change in size but not shape.
To further illustrate this, imagine you have a circle with radius 3 units and the center of dilation is located at the origin (0,0). If you dilate this circle by a scale factor of 0.5, each point on the circle will move towards the origin and the distance between the center and each point will be halved. As a result, the new circle will have a radius of 1.5 units, but its shape will remain circular, with all points equidistant from the center.
A dilation is not a rigid transformation because it changes the size of an object while preserving its proportions. The scale factor determines how much the object expands or contracts, and the center of dilation determines the direction and magnitude of the change. Unlike rigid transformations, which preserve both size and shape, dilation only preserves the shape of an object.