Question

rectangle lmnp was dilated using the rule $d_{p,3}$. which statements are true? choose three correct answers. the dilation is an enlargement. the scale factor is 3. the dilation is a reduction. the length of segment $mn$ is 14 units. the scale factor is $\frac{1}{3}$. the length of line segment $mn$ is 18 units.

Explanation:

Step1: Analyze Dilation Type

Dilation with scale factor \( k \): if \( k > 1 \), it's enlargement; if \( 0 < k < 1 \), it's reduction. Original rectangle LMNP has side lengths 4 (vertical) and 6 (horizontal). Dilated rectangle \( L'M'N'P' \) has vertical side 12.

Step2: Calculate Scale Factor

Scale factor \( k=\frac{\text{Image Length}}{\text{Original Length}}=\frac{12}{4} = 3 \). Since \( k = 3>1 \), dilation is enlargement.

Step3: Analyze \( M'N' \) Length

Original \( MN = 6 \), so \( M'N'=6\times3 = 18 \) units.

Step4: Evaluate Statements

• "The dilation is an enlargement": True ( \( k = 3>1 \) ).
• "The scale factor is 3": True (calculated \( k = 3 \) ).
• "The length of segment \( M'N' \) is 18 units": True ( \( 6\times3 = 18 \) ).
• "The dilation is a reduction": False ( \( k>1 \) ).
• "The scale factor is \( \frac{1}{3} \)": False ( \( k = 3 \) ).
• "The length of segment \( M'N' \) is 14 units": False ( \( 6\times3 = 18 \) ).

Answer:

The dilation is an enlargement, The scale factor is 3, The length of line segment \( M'N' \) is 18 units