In the study of geometry, a dilation is an important concept. A dilation is an enlargement or reduction of a figure with respect to a fixed point, called the center of dilation. This type of transformation is considered a nonrigid motion, as it does not preserve congruency. A dilation produces similar figures, where all corresponding angles are congruent, and all corresponding sides are proportional.
The scale factor indicates how much a figure will enlarge or reduce. The variable for scale factor is denoted as 'k', and it can be greater than 1 for an enlargement, equal to 1 for no change, or less than 1 for a reduction. When a point (x, y) undergoes a dilation with the origin (0, 0) as the center and a scale factor of 'k', the image of the point can be determined using the rule: (x, y) → (kx, ky).
Here are some examples of graphing and labeling figures and their images under a dilation with the given scale factor, assuming all dilations use the origin as the center of dilation:
- Triangle RST with vertices R(-5, 1), S(-3, 4), and 7(2,-1): k = 2
- Rectangle ABCD with vertices A(-3,0), B(1, 2), C(2, 0), and D(-2,-2): k = 3
- Rhombus JKLM with vertices J(-10, 2), K(-2, 8), L(6,2), and M(-2,-4): k = 1/2
- Triangle WXY with vertices W(-4, 8), X(10,0), and Y(-2,-8): k = 14
- Square DEFG with vertices D(0, 0), E(2, 4), F(6, 2), and G(4, -2): k= 5/2
- Triangle STU with vertices S(-15,-3), 7(-12, -15), and U(0, -9): k= 1/3
In order to understand dilations better, it is important to identify the scale factor given the graph of each image and its preimage. In addition, it is important to recognize the use of a scale factor when graphing the image of various shapes using a given scale factor.
In the study of geometry, a dilation is an important concept. A dilation is an enlargement or reduction of a figure with respect to a fixed point, called the center of dilation. This type of transformation is considered a nonrigid motion, as it does not preserve congruency. A dilation produces similar figures, where all corresponding angles are congruent, and all corresponding sides are proportional.
The scale factor indicates how much a figure will enlarge or reduce. The variable for scale factor is denoted as 'k', and it can be greater than 1 for an enlargement, equal to 1 for no change, or less than 1 for a reduction. When a point (x, y) undergoes a dilation with the origin (0, 0) as the center and a scale factor of 'k', the image of the point can be determined using the rule: (x, y) → (kx, ky).
Here are some examples of graphing and labeling figures and their images under a dilation with the given scale factor, assuming all dilations use the origin as the center of dilation:
- Triangle RST with vertices R(-5, 1), S(-3, 4), and 7(2,-1): k = 2
- Rectangle ABCD with vertices A(-3,0), B(1, 2), C(2, 0), and D(-2,-2): k = 3
- Rhombus JKLM with vertices J(-10, 2), K(-2, 8), L(6,2), and M(-2,-4): k = 1/2
- Triangle WXY with vertices W(-4, 8), X(10,0), and Y(-2,-8): k = 14
- Square DEFG with vertices D(0, 0), E(2, 4), F(6, 2), and G(4, -2): k= 5/2
- Triangle STU with vertices S(-15,-3), 7(-12, -15), and U(0, -9): k= 1/3
