Question

a triangle is dilated by a scale factor of ( n = \frac{1}{3} ). which statement is true regarding the dilation?
○ it is a reduction because ( n > 1 ).
○ it is a reduction because ( 0 < n < 1 ).
○ it is an enlargement because ( n > 1 ).
○ it is an enlargement because ( 0 > n > 1 ).

Explanation:

1. Recall the definition of dilation: A dilation with scale factor \( n \) is a reduction if \( 0 < n < 1 \) (the image is smaller than the original) and an enlargement if \( n>1 \) (the image is larger than the original).
2. Given the scale factor \( n=\frac{1}{3}\). We check the value of \( n \): \( 0<\frac{1}{3}< 1 \). So, according to the definition of dilation, when \( 0 < n < 1 \), the dilation is a reduction.
3. Now, analyze each option:

• Option 1: Says "reduction because \( n > 1 \)". But \( \frac{1}{3}<1 \), so this is wrong.
• Option 2: Says "reduction because \( 0 < n < 1 \)". Since \( 0<\frac{1}{3}<1 \), this is correct.
• Option 3: Says "enlargement because \( n > 1 \)". But \( \frac{1}{3}<1 \), so this is wrong.
• Option 4: The inequality \( 0>n > 1 \) is impossible (a number can't be both greater than 1 and less than 0), and also \( n=\frac{1}{3}\) is not in that range, so this is wrong.

Answer:

B. It is a reduction because \( 0 < n < 1 \) (where the option text is "It is a reduction because \( 0 < n < 1 \)")