Question

△rst is dilated with the rule ( d_{t, 1/3} (x, y) ), where the center of dilation is t(3, -2). the distance between the x-coordinates of r and t is checkmark. the distance between the y-coordinates of r and t is checkmark. r is dropdown from t, so the coordinates of r are dropdown with options (0, 0), (0, 2), (1, 2), (2, 0) (accompanied by a coordinate grid image of △rst with r, s, t plotted).

Explanation:

Step1: Find coordinates of R and T

From the graph, \( R(0, 4) \) and \( T(3, -2) \).

Step2: Distance between x - coordinates

The x - coordinate of \( R \) is \( 0 \), x - coordinate of \( T \) is \( 3 \). Distance \( = |3 - 0| = 3 \).

Step3: Distance between y - coordinates

The y - coordinate of \( R \) is \( 4 \), y - coordinate of \( T \) is \( - 2 \). Distance \( = |4-(-2)|=|4 + 2| = 6 \).

Step4: Dilation rule \( D_{T,1/3}(x,y) \)

Dilation with center \( T(3,-2) \) and scale factor \( \frac{1}{3} \). The vector from \( T \) to \( R \) is \( (0 - 3,4-(-2))=(-3,6) \). After dilation, the vector becomes \( \frac{1}{3}(-3,6)=(-1,2) \). Then \( R'=T+(-1,2)=(3-1,-2 + 2)=(2,0) \)? Wait, no, wait. Wait, the vector from \( T \) to \( R \) is \( R - T=(0 - 3,4-(-2))=(-3,6) \). After dilation by factor \( \frac{1}{3} \), the new vector is \( \frac{1}{3}(-3,6)=(-1,2) \). So \( R'=T+(-1,2)=(3-1,-2 + 2)=(2,0) \)? Wait, but let's check again. Wait, maybe I made a mistake. Wait, the distance between x - coordinates of \( R \) and \( T \) is \( |0 - 3| = 3 \), distance between y - coordinates is \( |4-(-2)| = 6 \). After dilation with scale factor \( \frac{1}{3} \), the distance from \( T \) to \( R' \) in x - direction is \( 3\times\frac{1}{3}=1 \) (since original distance is 3), and in y - direction is \( 6\times\frac{1}{3}=2 \). Since \( R \) is to the left and above \( T \), \( R' \) is \( 1 \) unit left and \( 2 \) units above \( T \). So \( x \) - coordinate of \( R' \): \( 3-1 = 2 \), \( y \) - coordinate: \( - 2+2 = 0 \). So \( R'=(2,0) \).

Answer:

• The distance between the x - coordinates of R and T is \( 3 \).
• The distance between the y - coordinates of R and T is \( 6 \).
• R' is \( \frac{1}{3} \) the distance from T (in terms of dilation), so the coordinates of R' are \( (2,0) \).