Question

the parallelogram rstu is a dilation of the parallelogram rstu. what is the scale factor of the dilation? simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Find coordinates of corresponding points

First, identify corresponding points of the original parallelogram \( RSTU \) and the dilated one \( R'S'T'U' \). Let's take point \( T \) and \( T' \). From the graph, \( T \) has coordinates \( (0, 1) \)? Wait, no, looking at the grid: \( T \) is at \( (0, 1) \)? Wait, no, let's check again. Wait, \( T \) in the original (green) is at \( (0, 1) \)? Wait, no, the green points: \( T \) is at \( (0, 1) \)? Wait, no, the y-coordinate for \( T \) (green) is 1? Wait, no, looking at the grid lines, each grid is 1 unit. Let's check \( T \) and \( T' \). \( T \) (original) is at \( (0, 1) \)? Wait, no, the green parallelogram: \( T \) is at (0,1)? Wait, no, \( R \) is at (0,2), \( T \) is at (0,1), \( S \) is at (-2,1), \( U \) is at (2,2)? Wait, no, maybe better to take the length of a side. Let's take the horizontal side from \( S \) to \( T \) in the original. \( S \) is at (-2,1), \( T \) is at (0,1), so the length is \( |0 - (-2)| = 2 \). In the dilated parallelogram \( S' \) to \( T' \): \( S' \) is at (-10,5)? Wait, no, \( S' \) is at (-10,5)? Wait, no, looking at the graph, \( S' \) is at (-10,5)? Wait, no, the blue parallelogram: \( T' \) is at (0,5)? Wait, no, \( T' \) is at (0,5)? Wait, \( R' \) is at (0,10), \( T' \) is at (0,5)? Wait, no, let's check coordinates properly.

Original parallelogram \( RSTU \): Let's find coordinates of \( T \) and \( T' \). \( T \) (original) is at (0, 1)? Wait, no, the green points: \( R \) is at (0, 2), \( T \) is at (0, 1), \( S \) is at (-2, 1), \( U \) is at (2, 2). So the vector from \( T \) to \( R \) is (0,1) (from (0,1) to (0,2)). In the dilated one, \( T' \) is at (0, 5)? Wait, no, \( T' \) is at (0, 5)? Wait, \( R' \) is at (0, 10), \( T' \) is at (0, 5). So the vector from \( T' \) to \( R' \) is (0,5) (from (0,5) to (0,10)). So the length of \( TR \) (original) is \( 2 - 1 = 1 \)? Wait, no, \( R \) is at (0,2), \( T \) is at (0,1), so the length is \( 2 - 1 = 1 \)? Wait, no, vertical distance? Wait, maybe horizontal side. \( S \) to \( T \): \( S \) is at (-2,1), \( T \) is at (0,1), so length is \( 0 - (-2) = 2 \). \( S' \) to \( T' \): \( S' \) is at (-10,5), \( T' \) is at (0,5), so length is \( 0 - (-10) = 10 \)? Wait, no, that can't be. Wait, maybe I messed up coordinates. Let's look again.

Wait, the original (green) parallelogram: \( S \) is at (-2,1), \( T \) is at (0,1), \( R \) is at (0,2), \( U \) is at (2,2). So the horizontal side \( ST \) has length \( 0 - (-2) = 2 \). The dilated (blue) parallelogram: \( S' \) is at (-10,5), \( T' \) is at (0,5), so length \( ST' \) (wait, \( S' \) to \( T' \)) is \( 0 - (-10) = 10 \)? No, that's too big. Wait, maybe \( T \) is at (0,1) and \( T' \) is at (0,5)? Wait, no, the blue \( T' \) is at (0,5)? Wait, the y-coordinate for \( T' \) is 5? Wait, the grid: each square is 1 unit. So \( T \) (green) is at (0,1), \( T' \) (blue) is at (0,5)? No, the blue \( T' \) is at (0,5)? Wait, \( R' \) is at (0,10), \( T' \) is at (0,5), so the vertical distance from \( T' \) to \( R' \) is \( 10 - 5 = 5 \). Original \( T \) to \( R \) is \( 2 - 1 = 1 \). So scale factor would be \( 5 / 1 = 5 \)? Wait, no, maybe horizontal side. Let's take \( U \) and \( U' \). \( U \) is at (2,2), \( U' \) is at (10,10). So the coordinates: \( U \) (2,2), \( U' \) (10,10). The change in x: \( 10 - 2 = 8 \), change in y: \( 10 - 2 = 8 \). Original vector from \( T \) to \( U \): \( T \) is (0,1), \( U \) is (2,2), so vector (2,1). Dilated vector from \( T' \) to \( U' \): \( T' \) is (0,5), \( U' \) is (10,10), so vector (10,5). So the scale factor is the ratio of the lengths. The length of \( TU \) (original) is \( \sqrt{(2)^2 + (1)^2} = \sqrt{5} \). The length of \( T'U' \) (dilated) is \( \sqrt{(10)^2 + (5)^2} = \sqrt{125} = 5\sqrt{5} \). So scale factor is \( 5\sqrt{5} / \sqrt{5} = 5 \). Alternatively, take the horizontal component: original \( TU \) has horizontal length 2 (from x=0 to x=2), dilated \( T'U' \) has horizontal length 10 (from x=0 to x=10). So scale factor is \( 10 / 2 = 5 \). Or vertical component: original \( TU \) has vertical length 1 (from y=1 to y=2), dilated \( T'U' \) has vertical length 5 (from y=5 to y=10). So scale factor is \( 5 / 1 = 5 \). So the scale factor is 5.

Step2: Confirm with another side

Let's check \( S \) to \( T \): original \( S \) is (-2,1), \( T \) is (0,1), length 2. Dilated \( S' \) is (-10,5), \( T' \) is (0,5), length 10. \( 10 / 2 = 5 \). So that confirms the scale factor is 5.

Answer:

5