When discussing the dilation of geometric shapes, such as triangles, the scale factor [tex]\( n \)[/tex] plays a critical role in determining whether the transformation results in an enlargement or a reduction of the shape.
To understand this, examine the scale factor:
1. If [tex]\( n > 1 \)[/tex], the original shape is enlarged because each dimension of the shape is multiplied by a factor greater than 1, making the shape bigger.
2. If [tex]\( 0 < n < 1 \)[/tex], the original shape is reduced because each dimension of the shape is multiplied by a factor less than 1 but greater than 0, making the shape smaller.
3. If [tex]\( n \leq 0 \)[/tex], the transformation involves a reflection or collapse, not a standard dilation applicable in this context.
Given that the scale factor for the triangle's dilation is [tex]\( n = \frac{1}{3} \)[/tex], we observe that:
- [tex]\( \frac{1}{3} \)[/tex] is less than 1 but greater than 0 (that is, [tex]\( 0 < \frac{1}{3} < 1 \)[/tex]).
Therefore, this scale factor indicates a reduction in size.
To determine the correct statement among the options provided:
1. "It is a reduction because [tex]\( n > 1 \)[/tex]." - This is incorrect because [tex]\( n > 1 \)[/tex] would indicate an enlargement, not a reduction.
2. "It is a reduction because [tex]\( 0 < n < 1 \)[/tex]." - This is correct because a scale factor between 0 and 1 results in a reduction in the size of the shape.
3. "It is an enlargement because [tex]\( n > 1 \)[/tex]." - This is incorrect because [tex]\( n = \frac{1}{3} \)[/tex] is less than 1, thus not leading to an enlargement.
4. "It is an enlargement because [tex]\( 0 > n > 1 \)[/tex]." - This is a nonsensical statement as it suggests [tex]\( n \)[/tex] can be both greater than 0 and greater than 1 simultaneously, and it incorrectly suggests an enlargement.
Thus, the correct statement regarding the dilation is:
"It is a reduction because [tex]\( 0 < n < 1 \)[/tex]."
