To determine which statement is true regarding the dilation of a triangle by a scale factor [tex]\( n = \frac{1}{3} \)[/tex], let's examine the given options step-by-step.
1. Understanding the Scale Factor:
A scale factor of [tex]\( n \)[/tex] determines how much each side of the triangle is multiplied by during the dilation process.
- If [tex]\( n > 1 \)[/tex], the triangle is enlarged (each side of the triangle increases in length).
- If [tex]\( 0 < n < 1 \)[/tex], the triangle is reduced (each side of the triangle decreases in length).
- If [tex]\( n < 0 \)[/tex], while we do not have such scenarios given here, it would generally involve a reflection and then a dilation based on magnitude.
Given [tex]\( n = \frac{1}{3} \)[/tex]:
- Notice that [tex]\(\frac{1}{3}\)[/tex] is a positive number.
- It is also less than 1 (since [tex]\(\frac{1}{3} < 1\)[/tex]).
2. Evaluating the Statements:
- Statement 1: It is a reduction because [tex]\( n > 1 \)[/tex].
- This statement is incorrect because [tex]\( n = \frac{1}{3} \)[/tex] is not greater than 1.
- Statement 2: It is a reduction because [tex]\( 0 < n < 1 \)[/tex].
- This statement is correct because [tex]\(\frac{1}{3}\)[/tex] lies between 0 and 1, indicating that the triangle is reduced in size.
- Statement 3: It is an enlargement because [tex]\( n > 1 \)[/tex].
- This statement is incorrect because [tex]\( n = \frac{1}{3} \)[/tex] is not greater than 1.
- Statement 4: It is an enlargement because [tex]\( 0 > n > 1 \)[/tex].
- This statement is incorrect because [tex]\(\frac{1}{3}\)[/tex] is not less than 0 nor greater than 1.
Therefore, the correct statement is:
It is a reduction because [tex]\( 0 < n < 1 \)[/tex].
