Verifying Dilations in the Coordinate Plane
Verifying dilations in the coordinate plane is a Grade 7 geometry skill in Big Ideas Math Advanced 2, Chapter 2: Transformations. To verify a dilation with center at the origin and scale factor k, check that each image coordinate equals k times the original coordinate and that the ratio of image to original coordinates is constant. If any ratio differs, the transformation is not a dilation.
Key Concepts
To verify a dilation with center at origin and scale factor $k$: 1. Check that each image coordinate equals $k$ times the original coordinate: $(x', y') = (kx, ky)$ 2. Confirm that the ratio of distances from origin is constant: $\frac{\text{distance to image point}}{\text{distance to original point}} = |k|$ 3. Verify that original and image points lie on the same ray from the origin.
Common Questions
How do you verify that a transformation is a dilation?
Divide each image coordinate by its corresponding original coordinate. If all ratios equal the same constant k, the transformation is a dilation with scale factor k centered at the origin.
How do you find the scale factor from coordinates?
Take any image point and divide its x-coordinate by the original x-coordinate (or y by y). For example, if A(2, 3) maps to A prime(6, 9), the scale factor is 6/2 equals 3 and 9/3 equals 3, so k equals 3.
What confirms that a transformation is not a dilation?
If the ratios of image to original coordinates are not all equal, the transformation is not a dilation. Different ratios for x and y indicate a different type of transformation.
What textbook covers verifying dilations in Grade 7?
Big Ideas Math Advanced 2, Chapter 2: Transformations covers how to verify dilations by checking coordinate ratios.