Question

quadrilateral qrst is dilated and translated to form similar figure qrst. what is the scale factor for the dilation?

Explanation:

Step1: Find length of QR and Q'R'

First, identify the length of a corresponding side in the original figure (QRST) and the dilated figure (Q'R'S'T). Let's take the horizontal side QR (or TS) and Q'R' (or T'S'). From the grid, QR spans from, say, x = -1 to x = 5? Wait, no, looking at the grid, Q'R' is from x = -1 to x = 1? Wait, no, the original figure QRST: let's check the coordinates. Wait, maybe better to count the number of grid units. Let's see, Q'R' is 2 units (from x=-1 to x=1, maybe? Wait, the small figure Q'R'S'T: the top side Q'R' is 2 units (since from x=-1 to x=1, that's 2 units). The original figure QRST: the bottom side QR (from x=-1 to x=5? Wait, no, looking at the x-axis, R is at x=5? Wait, no, the original figure's R is at x=5? Wait, no, the original figure (the larger one) has R at x=5? Wait, no, the small figure (Q'R'S'T) is above, and the large one is below. Wait, let's check the length of Q'R' and QR. Let's count the grid squares. Q'R' (top side of the small parallelogram) is 2 units (from x=-1 to x=1, so 2 units). The original figure's QR (top side? Wait, no, the original figure is below, with R at x=5? Wait, no, the original figure's R is at x=5? Wait, the original figure (QRST) has R at x=5, and Q at x=-1? Wait, no, the original figure's bottom side: from T (x=-3) to S (x=3)? Wait, maybe I'm overcomplicating. Let's take the length of Q'R' and QR. Wait, the small figure (Q'R'S'T) has Q'R' length 2 (horizontal), and the large figure (QRST) has QR length 6? Wait, no, wait the original figure: looking at the x-axis, R is at x=5, Q is at x=-1? No, the original figure's R is at x=5, and Q is at x=-1? Wait, no, the small figure is above, with Q' at x=-1, R' at x=1 (so length 2). The large figure: Q is at x=-1, R is at x=5? No, that can't be. Wait, maybe the original figure's QR is from x=-1 to x=3? Wait, no, the small figure's Q'R' is 2 units (from x=-1 to x=1), and the large figure's QR is 6 units? No, wait the large figure: from Q (x=-1) to R (x=5)? No, that's 6 units. Wait, no, let's look at the vertical sides. Wait, the small figure (Q'R'S'T) has a height of 2 (from y=0 to y=2), and the large figure (QRST) has a height of 6? No, wait the large figure's height: from y=0 to y=-6? Wait, no, the large figure is below the x-axis, from y=0 down to y=-6? Wait, no, the small figure is above, with height 2 (from y=0 to y=2), and the large figure is below, with height 6? No, that doesn't make sense. Wait, maybe the original figure's side length is 3 times the dilated one? Wait, no, let's check the length of Q'R' and QR. Wait, the small figure (Q'R'S'T) has Q'R' = 2 units (horizontal), and the original figure (QRST) has QR = 6 units? No, wait the original figure's QR: looking at the grid, from Q (x=-1) to R (x=5)? No, that's 6 units. Wait, no, the small figure is Q'R'S'T, with Q' at (-1,2), R' at (1,2), so Q'R' is 2 units (distance between x=-1 and x=1 is 2). The original figure: Q is at (-1,0), R is at (5,0)? No, that's 6 units. Wait, 2 and 6? No, that would be scale factor 1/3. Wait, no, maybe I got the figures reversed. Wait, the problem says "Quadrilateral QRST is dilated and translated to form similar figure Q'R'S'T". So QRST is the original, Q'R'S'T is the image. So we need to find the scale factor from QRST to Q'R'S'T. So length of Q'R' divided by length of QR. Let's find the length of Q'R' and QR. Let's take the horizontal side. Q'R' (image) length: from x=-1 to x=1, so 2 units. QR (original) length: from x=-1 to x=5? No, wait the original figure's R is at x=5? Wait, no, the original figure (QRST) has R at x=5, Q at x=-1? No, the original figure's bottom side: from T (x=-3) to S (x=3), and Q (x=-1) to R (x=3)? Wait, no, the original figure is a parallelogram. Wait, maybe the original figure's QR is 6 units, and the image's Q'R' is 2 units? Then scale factor is 2/6 = 1/3? Wait, no, wait the original figure: looking at the x-axis, the original figure's R is at x=5, Q is at x=-1? No, the original figure's R is at x=5, Q is at x=-1, so QR length is 6. The image's Q'R' is from x=-1 to x=1, length 2. So scale factor is 2/6 = 1/3? Wait, no, maybe I made a mistake. Wait, let's check the vertical sides. The small figure (Q'R'S'T) has a height of 2 (from y=0 to y=2), and the large figure (QRST) has a height of 6 (from y=0 to y=-6)? No, that would be 2/6 = 1/3. Wait, but maybe the original figure's length is 3 times the image? Wait, no, the scale factor is image length / original length. So if original is QRST, image is Q'R'S'T, then scale factor is length of Q'R' / length of QR. Let's count the grid squares. Q'R' is 2 units (from x=-1 to x=1, 2 units). QR: from Q (x=-1) to R (x=5)? No, that's 6 units. Wait, 2/6 = 1/3? Wait, no, maybe the original figure's QR is 3 units? Wait, no, the original figure: looking at the x-axis, the original figure's R is at x=3? Wait, no, the original figure's R is at x=5? Wait, the grid has x-axis with -4, -2, 0, 2, 4, 6? Wait, the original figure's R is at x=5? No, the grid lines are at integers: -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6. So Q' is at (-1,2), R' at (1,2), so Q'R' is 2 units (from x=-1 to x=1, distance 2). The original figure's Q is at (-1,0), R at (5,0)? No, that's 6 units. Wait, maybe the original figure's QR is 3 units? Wait, no, the original figure: looking at the bottom side, from T (x=-3) to S (x=3), so length 6, and the top side QR is also 6. The image's T'S' is 2 units, so scale factor is 2/6 = 1/3. Wait, but let's check another side. The small figure's height (vertical) is 2 (from y=0 to y=2), and the large figure's height is 6 (from y=0 to y=-6), so 2/6 = 1/3. So the scale factor is 1/3? Wait, no, wait the problem says "dilated and translated". Dilation scale factor: if the image is smaller, scale factor is less than 1. Wait, but maybe I got the figures reversed. Wait, the original figure is Q'R'S'T and the image is QRST? No, the problem says "Quadrilateral QRST is dilated and translated to form similar figure Q'R'S'T". So QRST is original, Q'R'S'T is image. So image is smaller, so scale factor is image length / original length. Let's confirm with coordinates. Let's take point Q: in QRST, Q is at (-1, 0). In Q'R'S'T, Q' is at (-1, 2). Wait, no, translation is involved, so we need to find corresponding sides. Let's take the length of Q'R' and QR. Q'R' (image) length: from x=-1 to x=1 (2 units). QR (original) length: from x=-1 to x=5? No, that's 6 units. Wait, 2/6 = 1/3. Wait, but maybe the original figure's QR is 3 units? Wait, no, the original figure's R is at x=5? Wait, the original figure's R is at x=5, Q at x=-1, so distance is 6. The image's Q' is at x=-1, R' at x=1, distance 2. So 2/6 = 1/3. So scale factor is 1/3? Wait, no, wait the original figure: looking at the bottom side, from T (x=-3) to S (x=3), that's 6 units. The image's T' is at x=-2, S' at x=1? No, T' is at x=-2, S' at x=1? Wait, no, T' is at x=-2, Q' at x=-1, R' at x=1, S' at x=0? Wait, maybe I messed up the coordinates. Let's start over.

Looking at the grid:

- Small figure (Q'R'S'T): Q' is at (-1, 2), R' at (1, 2), S' at (0, 0), T' at (-2, 0). So Q'R' is from (-1,2) to (1,2): length is 2 (horizontal distance: 1 - (-1) = 2).

- Original figure (QRST): Q is at (-1, 0), R at (5, 0), S at (3, -6), T at (-3, -6). So QR is from (-1,0) to (5,0): length is 5 - (-1) = 6.

So scale factor is length of Q'R' / length of QR = 2 / 6 = 1/3.

Step2: Confirm scale factor

So the scale factor is the ratio of the length of a side in the image (Q'R'S'T) to the corresponding side in the original (QRST). We found Q'R' = 2, QR = 6, so 2/6 = 1/3.

Answer:

$\frac{1}{3}$