Transformation in geometry

Dilation is a transformation in geometry that proportionally resizes a figure while preserving its shape

Non-isometric transformation

Changes in size and proportions

Dilation is a non-isometric transformation that changes the size of a figure while maintaining its proportions and angles

Center of dilation

Dilation occurs with respect to a fixed point known as the center of dilation

Scale factor

Dilation involves a scale factor that determines the degree of resizing

Effects of Dilation

Enlargement and reduction

A scale factor greater than one causes an enlargement, while a scale factor between zero and one results in a reduction

Congruence

A scale factor of one implies that the figure remains unchanged, as the image is congruent to the pre-image

Retention of shape

Dilation retains the intrinsic shape of a figure, preserving angles, parallelism, and perpendicularity

Proportionality

Proportionality of segments

Dilation maintains the proportionality of segments, where the ratio of lengths of corresponding segments in the pre-image and image is constant and equal to the scale factor

Preservation of midpoints

Dilation preserves the midpoints of segments, where the midpoint of any segment in the dilated figure corresponds to the midpoint of the related segment in the pre-image

Determining Dilation

The scale factor is the key to performing a dilation, and it can be determined by comparing corresponding points or segments in the pre-image and image

Dilation with origin as center

When the origin is the center of dilation, the coordinates of the pre-image are directly scaled by the scale factor to determine the coordinates of the image

Dilation with non-origin center

Calculating vectors

When the center of dilation is not at the origin, one must calculate the vectors from the center to the vertices of the pre-image

Finding new coordinates

These vectors are then multiplied by the scale factor to determine the corresponding vectors for the image, and the endpoints of these new vectors give the coordinates of the vertices of the dilated figure

Practical uses

Dilation has practical applications in geometry, such as scale models and real-world contexts

Examples

Enlargement and reduction

A square with a scale factor of 2 will result in a figure twice the size, while a triangle with a scale factor of 0.5 will be half the size

Negative scale factor

A negative scale factor results in both resizing and reflection across the center of dilation

Determining scale factor

The scale factor can be determined by comparing corresponding points or segments in the pre-image and image