Question

1. 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)→(ax, ay). which statement must not be true? if a is greater than 1, the image of the polygon is larger than the original polygon. if a is between 0 and 1, the image of the polygon is smaller than the original polygon. if a is greater than 1, the image of the polygon is smaller than the original polygon. if a is equal to 1, the image of the polygon is congruent to the original polygon.

Explanation:

Step1: Understand dilation concept

The rule $(x,y)\to(ax,ay)$ represents a dilation. When $a > 1$, it is an enlargement. When $0 < a<1$, it is a reduction. When $a = 1$, the figure is unchanged.

Step2: Analyze each option

• Option 1: If $a>1$, multiplying the coordinates by $a$ makes the distances from the origin larger, so the image is larger than the original. This is true.
• Option 2: If $0 < a<1$, multiplying the coordinates by $a$ makes the distances from the origin smaller, so the image is smaller than the original. This is true.
• Option 3: If $a>1$, the image cannot be smaller than the original. This is false.
• Option 4: If $a = 1$, $(x,y)\to(1x,1y)=(x,y)$, so the image is congruent to the original. This is true.

Answer:

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