Question

line segment qr is dilated to create line segment qr using the dilation rule d_{r,1.5}. what is y, the distance between points r and r? 3 units 6 units 4 units 9 units

Explanation:

Step1: Understand Dilation Scale

The dilation rule is \( D_{r,1.5} \), so the scale factor \( r = 1.5 \). The length \( TR = 6 \) units. After dilation, \( TR' = TR \times 1.5 = 6 \times 1.5 = 9 \) units.

Step2: Calculate \( y \) (Distance \( R \) to \( R' \))

We know \( TR' = TR + y \). Substitute \( TR' = 9 \) and \( TR = 6 \): \( y = TR' - TR = 9 - 6 = 3 \)? Wait, no—wait, maybe I misread. Wait, the dilation is of \( QR \) to \( Q'R' \), but \( TR \) is along the same line. Wait, actually, the scale factor is 1.5, so \( TR' = 1.5 \times TR \). \( TR = 6 \), so \( TR' = 9 \). Then \( y = TR' - TR = 9 - 6 = 3 \)? But wait, the options have 9,4,6,3. Wait, no—wait, maybe \( TR \) is 6, and \( TR' \) is 6*1.5=9, so the distance between R and R' is \( 9 - 6 = 3 \)? Wait, no, maybe I messed up. Wait, the dilation center is T? So \( TQ' = 1.5 \times TQ \), and \( TR' = 1.5 \times TR \). So \( TR = 6 \), \( TR' = 6 \times 1.5 = 9 \). Then \( y = TR' - TR = 9 - 6 = 3 \)? But the first option is 9, second 4, third 6, fourth 3. Wait, maybe I made a mistake. Wait, no—wait, the problem is asking for the distance between R and R', which is \( TR' - TR \). Since \( TR' = 1.5 \times TR = 1.5 \times 6 = 9 \), then \( y = 9 - 6 = 3 \)? But the option with 3 is there. Wait, but let me check again. Wait, maybe the scale factor is 1.5, so \( TR' = 1.5 \times TR \), so \( TR' = 9 \), so the distance from R to R' is \( 9 - 6 = 3 \). So the answer should be 3 units? Wait, no, wait the options: first is 9, second 4, third 6, fourth 3. Wait, maybe I misread the length of TR. Wait, the diagram shows TR as 6, then R' is further. So \( TR' = 6 \times 1.5 = 9 \), so R to R' is \( 9 - 6 = 3 \). So the answer is 3? Wait, but the first option is 9, which is TR'. Wait, no, the question is "the distance between points R and R'". So \( TR' - TR = 9 - 6 = 3 \). So the answer is 3 units.

Answer:

3 units (the option with "3 units")