Question

dilating a figure with the center of dilation at a point
△rst is dilated with the rule ( d_{t, \frac{1}{2}} (x, y) ), where the center of dilation is t (3, -2).
the distance between the x-coordinates of r and t is
the distance between the y-coordinates of r and t is
r is from t, so the coordinates of r are
(0, 0)
(0, 2)
(1, 2)
(2, 0)

Explanation:

Step1: Find coordinates of R and T

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

Step2: Distance between x - coordinates

\( |0 - 3| = 3 \)? Wait, no, wait. Wait, R is at x = 0, T at x = 3? Wait, no, looking at the graph, R is at (0,4), T is at (3, -2). Wait, the x - coordinate of R is 0, x - coordinate of T is 3. Wait, but maybe I misread. Wait, the grid: S is at (0, -2)? Wait, no, S is at (0, -2)? Wait, no, the graph: R is at (0,4), S at (0, -2)? No, wait, the y - axis: R is at y = 4, S at y = -2? No, the shaded triangle: R is at (0,4), S at (0, -2)? No, T is at (3, -2). Wait, x - coordinate of R: 0, x - coordinate of T: 3. So distance between x - coordinates: \( |0 - 3| = 3 \)? Wait, but the dilation factor is \( \frac{1}{3} \)? Wait, no, the rule is \( D_{T,\frac{1}{3}}(x,y) \). So first, find coordinates of R: from the graph, R is (0,4), T is (3, -2).

Distance between x - coordinates of R and T: \( |0 - 3| = 3 \).

Distance between y - coordinates of R and T: \( |4 - (-2)| = |4 + 2| = 6 \).

Now, dilation with center T and scale factor \( \frac{1}{3} \). So the vector from T to R is \( (0 - 3, 4 - (-2)) = (-3, 6) \). Multiply by scale factor \( \frac{1}{3} \): \( (-3\times\frac{1}{3}, 6\times\frac{1}{3}) = (-1, 2) \). Then add to T's coordinates: \( (3 + (-1), -2 + 2) = (2, 0) \)? Wait, no, wait. Wait, maybe I messed up the direction. The center is T, so to find R', we go from T towards R, scaled by \( \frac{1}{3} \). So the x - distance from T to R: R is at x = 0, T at x = 3. So the difference is \( 0 - 3 = -3 \). Scaled by \( \frac{1}{3} \): \( -3\times\frac{1}{3} = -1 \). So new x - coordinate: \( 3 + (-1) = 2 \). Y - distance: R at y = 4, T at y = -2. Difference: \( 4 - (-2) = 6 \). Scaled by \( \frac{1}{3} \): \( 6\times\frac{1}{3} = 2 \). New y - coordinate: \( -2 + 2 = 0 \). So R' is (2, 0). Wait, but the options include (2,0). Wait, but first, the distance between x - coordinates of R and T: R is at x = 0, T at x = 3. So distance is \( |0 - 3| = 3 \). Wait, but maybe the dilation factor is \( \frac{1}{3} \), so the distance from T to R' in x - direction is \( 3\times\frac{1}{3}=1 \)? Wait, no, original distance between x - coordinates (R to T) is 3, so after dilation, the distance from T to R' in x - direction is \( 3\times\frac{1}{3}=1 \)? Wait, no, R is to the left of T (x = 0 < x = 3), so the distance between x - coordinates is 3 (T is at x = 3, R at x = 0, so 3 units left). After dilation with scale factor \( \frac{1}{3} \), the distance from T to R' in x - direction is \( 3\times\frac{1}{3}=1 \) unit left. So x - coordinate of R': \( 3 - 1 = 2 \).

Distance between y - coordinates: R is at y = 4, T at y = -2, so 6 units up. After dilation, distance from T to R' in y - direction is \( 6\times\frac{1}{3}=2 \) units up. So y - coordinate of R': \( -2 + 2 = 0 \). So R' is (2, 0).

Wait, but the first two blanks: distance between x - coordinates of R and T: 3? Wait, no, maybe I misread R's x - coordinate. Wait, looking at the graph, R is at (0,4), T is at (3, -2). So x - coordinate of R: 0, x - coordinate of T: 3. So distance is \( |0 - 3| = 3 \). Distance between y - coordinates: \( |4 - (-2)| = 6 \).

Then, for R', the coordinates: as above, (2,0).

So first blank: 3? Wait, but maybe the graph has R at (0,4), T at (3, -2). So x - distance: 3, y - distance: 6. Then R' is (2,0).

Answer:

First blank: 3, Second blank: 6, Third blank: (2, 0)