Question

line segment su is dilated to create su using the dilation rule ( d_{q,2.5} ). what is the distance, ( x ), between points u and u?
options: 10 units, 4.8 units, 4 units, 6 units

Explanation:

Step1: Find the scale factor

The dilation rule is \( D_{Q,1.5} \), so the scale factor \( k = 1.5 \). We can also verify this using the segments from \( Q \) to \( S \) and \( Q \) to \( S' \). The length \( QS = 3.2 \) and \( QS' = 3.2 + 4.8 = 8 \)? Wait, no, actually, dilation scales the distance from the center \( Q \). So \( QS' = k \times QS \). Let's check: \( QS = 3.2 \), \( QS' = 3.2 + 4.8 = 8 \)? Wait, no, maybe \( QS = 3.2 \), and \( QS' = 3.2 \times 1.5 = 4.8 \)? Wait, no, the diagram shows \( QS = 3.2 \), \( SS' = 4.8 \), so \( QS' = QS + SS' = 3.2 + 4.8 = 8 \). Then \( 3.2 \times k = 8 \), so \( k = 8 / 3.2 = 2.5 \)? Wait, no, the dilation rule is \( D_{Q,1.5} \), so scale factor is 1.5. Wait, maybe I misread. Let's look at \( QU \) and \( QU' \). \( QU = 4 \), so \( QU' = 4 \times 1.5 = 6 \). Then the distance between \( U \) and \( U' \) is \( QU' - QU = 6 - 4 = 2 \)? No, that's not matching. Wait, maybe the scale factor is calculated from \( QS \) and \( QS' \). \( QS = 3.2 \), \( QS' = 3.2 + 4.8 = 8 \)? No, 3.2 * 2.5 = 8. But the dilation rule is \( D_{Q,1.5} \)? Wait, maybe the problem says dilation rule \( D_{Q,1.5} \), so scale factor \( k = 1.5 \). Then \( QU' = k \times QU = 1.5 \times 4 = 6 \). Then the distance between \( U \) and \( U' \) is \( QU' - QU = 6 - 4 = 2 \)? No, that's not one of the options. Wait, maybe I made a mistake. Wait, the options are 6, 4, 4.8, 10? Wait, no, the question is the distance between \( U' \) and \( U \). Wait, \( QU = 4 \), \( QU' = 4 \times 1.5 = 6 \), so \( UU' = QU' - QU = 6 - 4 = 2 \)? No, that's not an option. Wait, maybe the scale factor is from \( QS \) to \( QS' \). \( QS = 3.2 \), \( QS' = 3.2 + 4.8 = 8 \), so scale factor \( k = 8 / 3.2 = 2.5 \). Then \( QU' = 4 \times 2.5 = 10 \), so \( UU' = 10 - 4 = 6 \). Ah, that's one of the options (6 units). So let's redo:

Step1: Determine the scale factor

Using the segments from \( Q \) to \( S \) and \( Q \) to \( S' \). \( QS = 3.2 \), \( QS' = 3.2 + 4.8 = 8 \). The scale factor \( k = \frac{QS'}{QS} = \frac{8}{3.2} = 2.5 \). Wait, but the dilation rule is \( D_{Q,1.5} \)? Maybe the diagram's labels are different. Alternatively, maybe \( QU = 4 \), and after dilation, \( QU' = 4 + x \), and since dilation scales the length from \( Q \), \( QU' = k \times QU \). If we use the \( S \) segment: \( QS = 3.2 \), \( QS' = 3.2 + 4.8 = 8 \), so \( k = 8 / 3.2 = 2.5 \). Then \( QU' = 4 \times 2.5 = 10 \), so \( x = QU' - QU = 10 - 4 = 6 \).

Step2: Calculate \( x \)

Using the scale factor \( k = 2.5 \) (from \( QS \) and \( QS' \)), the length \( QU' = k \times QU = 2.5 \times 4 = 10 \). Then the distance between \( U \) and \( U' \) is \( QU' - QU = 10 - 4 = 6 \).

Answer:

6 units (the option with 6 units)