Question

dilating a figure with the center of dilation at a point
△rst is dilated with the rule ( d_{t, \frac{1}{3}} (x, y) ), where the center of dilation is t (3, -2).
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 options: 1 unit left, 1 unit up; 1 unit left, 2 units up; 3 units left, 2 units up; 2 units left, 4 units up from t, so the coordinates of r are 2...

Explanation:

Step1: Find coordinates of R

From the graph, R is at (0, 4) (since it's on the y - axis, x = 0, y = 4). T is at (3, - 2).

Step2: Calculate x - distance between R and T

The x - coordinate of R is 0, x - coordinate of T is 3. The distance between them is $|3 - 0|=3$.

Step3: Calculate y - distance between R and T

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

Step4: Analyze dilation rule

The dilation rule is $D_{T,\frac{1}{3}}(x,y)$, which means we scale the distance from T to R by a factor of $\frac{1}{3}$.
The x - distance from T to R is 3 (from T(3,-2) to R(0,4), the horizontal change is $0 - 3=-3$, absolute value 3). Scaling by $\frac{1}{3}$: $3\times\frac{1}{3}=1$. So R' is 1 unit left from T (since R is to the left of T, the direction is left).
The y - distance from T to R is 6 (vertical change is $4-(-2) = 6$). Scaling by $\frac{1}{3}$: $6\times\frac{1}{3}=2$. So R' is 2 units up from T (since R is above T, the direction is up).
So R' is 1 unit left, 2 units up from T.

Answer:

1 unit left, 2 units up