Question

AB determine whether the dilation passes through the center of dilation

Answer 1

To determine whether a dilation passes through the center of dilation, we need to understand what dilation is in the context of geometry.

A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. This transformation involves a center of dilation and a scale factor.

1. **Center of Dilation**: This is a fixed point in the plane (usually denoted as O). All points are expanded or contracted from this center based on the scale factor.

2. **Scale Factor**: This is a positive number that determines how much larger or smaller the image will be compared to the original shape. A scale factor greater than 1 enlarges the figure, while a scale factor between 0 and 1 reduces it.

For a point A being dilated to A', by the center O and scale factor k, the following relationship holds:

- A' is located by using the formula:

\[
A' = O + k(A - O)
\]

This means that the point A' is determined by a vector that originates at the center O and points towards A, scaled by the factor k.

### Analysis:

To see if the dilation "passes through" the center of dilation (point O), we need to check the following:

- When **A** is equal to **O**, the dilation would yield:

\[
A' = O + k(O - O) = O + k(0) = O
\]

So, if the original point A is the same as the center of dilation O, then the dilated image A' will also be at O.

- When **A** is **not** equal to O (meaning A is any point other than O), A' will not be O unless k = 1 (which keeps point A at O), and the point will move away from O depending on the scale factor.

### Conclusion:

- A dilation passes through the center of dilation if and only if at least one point being dilated is the center of dilation, or the scale factor is specifically 1 (which means the dilation does not change the size of the figure). Otherwise, generally, the image produced by the dilation will not pass through the center.