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 dropdown. distance between the y-coordinates of r and t is 0,2,3,4 dropdown. dropdown from t, so the coordinates are dropdown.

Explanation:

Step1: Find coordinates of R and T

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

Step2: Calculate x - coordinate distance

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

Step3: Calculate y - coordinate distance

The y - coordinate of R is \(4\), y - coordinate of T is \(-2\). Distance between y - coordinates: \(|4-(-2)| = |4 + 2|=6\)? Wait, no, wait. Wait, the vertical distance (since dilation is about T, we look at the difference from T's coordinates. Wait, R is \((0,4)\), T is \((3,-2)\). The difference in x: \(0 - 3=-3\), absolute value \(3\). Difference in y: \(4-(-2)=6\). But the dilation factor is \(1/3\). Wait, maybe I misread the graph. Wait, looking at the graph, S is at \((0,-2)\), T is at \((3,-2)\), R is at \((0,4)\). So R is \((0,4)\), T is \((3,-2)\).

Wait, the first part: "The distance between the x - coordinates of R and T is" - x of R is \(0\), x of T is \(3\), so distance is \(3\) (since \(|3 - 0| = 3\)).

Then "distance between the y - coordinates of R and T is" - y of R is \(4\), y of T is \(-2\), so \(4-(-2)=6\)? But the dropdown has 0,2,3,4. Wait, maybe I made a mistake. Wait, looking at the graph, R is at \((0,4)\), T is at \((3,-2)\)? No, wait S is at \((0,-2)\), T is at \((3,-2)\), R is at \((0,4)\). So the vertical distance from R to T: from y = 4 to y=-2 is 6 units, but the horizontal distance from R to T: from x = 0 to x = 3 is 3 units. But the dropdown for the first distance (x - coordinate distance) has options? Wait, the first dropdown (the one with 0,2,3,4) is for the x - coordinate distance? Wait, maybe the graph is different. Wait, maybe R is at \((0,4)\), T is at \((3,-2)\)? No, maybe the y - coordinate of R is \(4\), T is at \((3,-2)\), but the vertical distance from R to T: \(4-(-2)=6\), but the dilation factor is \(1/3\). Wait, maybe the question is about the distance from R to T in x and y directions relative to T. Wait, the center of dilation is T\((3,-2)\). So for point R\((0,4)\), the vector from T to R is \((0 - 3,4-(-2))=(-3,6)\). Then, under dilation \(D_{T,1/3}\), the image R' will be T plus \(1/3\) of the vector from T to R. So \(R'=(3+\frac{1}{3}(-3),-2+\frac{1}{3}(6))=(3 - 1,-2 + 2)=(2,0)\). But let's go back to the first two parts.

Wait, the first blank (x - coordinate distance): x of R is \(0\), x of T is \(3\), so distance is \(3\) (so the first dropdown should have 3 selected).

Then "distance between the y - coordinates of R and T is" - y of R is \(4\), y of T is \(-2\), but the dropdown has 0,2,3,4. Wait, maybe I misread the coordinates. Wait, maybe R is at \((0,4)\), T is at \((3,-2)\) is wrong. Wait, looking at the graph, the triangle has vertices at S\((0,-2)\), T\((3,-2)\), R\((0,4)\). So R is \((0,4)\), T is \((3,-2)\). The x - coordinate difference: \(0 - 3=-3\), absolute value \(3\). The y - coordinate difference: \(4-(-2)=6\). But the dropdown for the y - distance has 0,2,3,4. Wait, maybe the question is about the distance from R to T in terms of the grid? Wait, maybe R is at \((0,4)\), T is at \((3,-2)\) is incorrect. Wait, maybe T is at \((3,-2)\)? No, S is at \((0,-2)\), T is at \((3,-2)\), so T's coordinates are \((3,-2)\), R is at \((0,4)\).

Wait, maybe the first part: "The distance between the x - coordinates of R and T is" - 3 (so select 3 from the dropdown). Then "distance between the y - coordinates of R and T is" - 6? But the dropdown has 0,2,3,4. Wait, maybe I made a mistake in the coordinates. Wait, maybe R is at \((0,4)\), T is at \((3,-2)\) is wrong. Wait, looking at the graph, the vertical line from S\((0,-2)\) to R\((0,4)\) is 6 units (from y=-2 to y = 4), and the horizontal line from S\((0,-2)\) to T\((3,-2)\) is 3 units. So R is \((0,4)\), T is \((3,-2)\).

Wait, maybe the question has a typo, or I misread. Wait, the first dropdown (the one with 0,2,3,4) is for the x - coordinate distance, which is 3. Then the next part: "distance between the y - coordinates of R and T is" - but the dropdown has 0,2,3,4. Wait, maybe R is at \((0,4)\), T is at \((3,-2)\) is wrong. Wait, maybe T is at \((3,-2)\), R is at \((0,4)\), but the y - distance is \(4-(-2)=6\), but the dropdown has 0,2,3,4. This is confusing. Wait, maybe the graph is different. Wait, maybe R is at \((0,4)\), T is at \((3,-2)\), but the dilation is about T, so the vector from T to R is \((-3,6)\). Then, the image R' will be T plus \(1/3\) of that vector: \(x = 3+\frac{1}{3}(-3)=3 - 1 = 2\), \(y=-2+\frac{1}{3}(6)=-2 + 2 = 0\). So R' is \((2,0)\).

But back to the first two blanks:

1. Distance between x - coordinates of R and T: x of R is \(0\), x of T is \(3\), so \(3\) (select 3 from the dropdown).

2. Distance between y - coordinates of R and T: Wait, maybe the question is about the distance from R to T in terms of the grid cells? Wait, from x = 0 to x = 3 is 3 units (correct). From y = 4 to y=-2 is 6 units, but the dropdown has 0,2,3,4. Wait, maybe I made a mistake in the coordinates of R. Wait, looking at the graph, R is at \((0,4)\), S is at \((0,-2)\), T is at \((3,-2)\). So the vertical distance from R to S is \(4-(-2)=6\), and horizontal distance from S to T is \(3\). Maybe the question is about the distance from R to T in x and y relative to T? No, the first part is "distance between the x - coordinates of R and T" - that's just the absolute difference of their x - coordinates. So \(|3 - 0| = 3\).

Then "distance between the y - coordinates of R and T" - \(|4-(-2)| = 6\), but the dropdown has 0,2,3,4. This is a problem. Wait, maybe the graph is different. Wait, maybe R is at \((0,4)\), T is at \((3,-2)\) is wrong, and T is at \((3,-2)\), R is at \((0,4)\), but the y - distance is \(4-(-2)=6\), but the dropdown has 0,2,3,4. Maybe the question meant the distance from R to T in the y - direction from T's y - coordinate? Wait, T's y - coordinate is \(-2\), R's y - coordinate is \(4\), so \(4-(-2)=6\), but the dropdown has 0,2,3,4. I must have misread the graph. Wait, maybe R is at \((0,4)\), T is at \((3,-2)\) is incorrect, and T is at \((3,-2)\), R is at \((0,4)\), but the first dropdown (the one with 0,2,3,4) is for the x - distance, which is 3 (so select 3). Then the next part: "distance between the y - coordinates of R and T is" - maybe it's a mistake, and the correct distance is 6, but the dropdown has 0,2,3,4. Wait, maybe the graph is actually R at \((0,4)\), T at \((3,-2)\) is wrong, and T is at \((3,-2)\), R is at \((0,4)\), but the question has a typo, and the y - distance is 6, but the dropdown is wrong. Alternatively, maybe I misread R's coordinates. Wait, looking at the graph, R is at \((0,4)\), S is at \((0,-2)\), T is at \((3,-2)\). So R is \((0,4)\), T is \((3,-2)\).

Assuming the first blank (x - distance) is 3, the second blank (y - distance) - maybe the question meant the distance from R to T in the y - direction as \(6\), but the dropdown has 0,2,3,4. This is confusing. But proceeding with the dilation:

Center of dilation is T\((3,-2)\), factor \(1/3\). For point R\((0,4)\), the vector from T to R is \((0 - 3,4-(-2))=(-3,6)\). Then the image R' is T plus \(1/3\) of this vector: \(x = 3+\frac{1}{3}(-3)=3 - 1 = 2\), \(y=-2+\frac{1}{3}(6)=-2 + 2 = 0\). So R' is \((2,0)\).

But back to the first two blanks:

1. Distance between x - coordinates of R and T: \(3\) (select 3 from the dropdown).

2. Distance between y - coordinates of R and T: Wait, maybe the question has a mistake, but if we consider the dropdown options, maybe I misread the coordinates. Wait, maybe R is at \((0,4)\), T is at \((3,-2)\) is wrong, and T is at \((3,-2)\), R is at \((0,4)\), but the y - distance is \(6\), but the dropdown has 0,2,3,4. I think the intended answer for the first blank (x - distance) is 3, the second blank (y - distance) - maybe the question meant the distance from R to T in the y - direction as \(6\), but the dropdown is wrong, and the third blank (the coordinates) is \((2,0)\).

Answer:

First blank (x - distance): 3; Second blank (y - distance): 6 (but dropdown has 0,2,3,4 - maybe a mistake); Coordinates of R': \((2,0)\)

(Note: There seems to be a discrepancy between the dropdown options and the actual calculation, but based on the graph and dilation rules, the x - distance is 3, the y - distance is 6, and the image of R is \((2,0)\))