A polygon is graphed on a coordinate plane with (x, y) representing the location of a certain point on the polygon. The polygon is transformed using the rule (x, y)  symbol (ax, ay).
 
Which statement must NOT be true?
A
If a is greater than 1, the image of the polygon is larger than the original polygon.
B
If a is between 0 and 1, the image of the polygon is smaller than the original polygon.
C
If a is greater than 1, the image of the polygon is smaller than the original polygon.
D
If a is equal to 1, the image of the polygon is congruent to the original polygon.

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!

A polygon is graphed on a coordinate plane with (x, y) representing the location of a certain point on the polygon. The polygon is transformed using the rule (x, y) symbol (ax, ay). Which statement must NOT be true? A) If a is greater than 1, the image of the polygon is larger than the original polygon. B) If a is between 0 and 1, the image of the polygon is smaller than the original polygon. C) If a is greater than 1, the image of the polygon is smaller than the original polygon. D) If a is equal to 1, the image of the polygon is congruent to the original polygon.

Explore the effects of dilation on polygons on a coordinate plane, focusing on how the scale factor 'a' transforms the image. Learn which conditions make the polygon larger, smaller, or congruent to the original. This problem is suitable for students in Grades 9-12 and covers key geometric concepts of dilation and scaling.

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