To analyze this question, we are dealing with a transformation rule where each point $(x, y)$ of a polygon is transformed into $(ax, ay)$, based on the constant $a$. This transformation represents a dilation or scaling of the polygon, depending on the value of $a$.

Let's examine the options:

A. If $a > 1$, the image of the polygon is larger than the original polygon.

• This statement is true because when $a > 1$, each coordinate $(x, y)$ is multiplied by a number greater than 1, which stretches the polygon outward, making the image larger than the original.

B. If $0 < a < 1$, the image of the polygon is smaller than the original polygon.

• This statement is true because when $a$ is between 0 and 1, each coordinate $(x, y)$ is multiplied by a fraction, which compresses the polygon, making it smaller than the original.

C. If $a > 1$, the image of the polygon is smaller than the original polygon.

• This statement is false because, as already explained in option A, when $a > 1$, the polygon becomes larger, not smaller.

D. If $a = 1$, the image of the polygon is congruent to the original polygon.

• This statement is true because when $a = 1$, each point $(x, y)$ stays the same, so the polygon does not change in size or shape and is congruent to the original.

Conclusion:

The statement that must NOT be true is C.

Would you like further details or clarifications?

Here are some related questions:

1. What happens when $a$ is negative in a transformation?
2. How does the area of a polygon change with this kind of dilation?
3. What is the geometric interpretation of dilations in a coordinate plane?
4. How does the concept of similarity relate to dilation transformations?
5. What other transformations can be combined with dilation to change the shape of a polygon?

Tip: When analyzing transformations, always check how the scale factor affects size, orientation, and congruence!