Question

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

Explanation:

Step1: Find length of Q'R'

From the graph, Q'R' spans 2 units (e.g., from x=-1 to x=1, or visually 2 grid units).

Step2: Find length of QR

QR spans 6 units (e.g., from x=-1 to x=5, or visually 6 grid units? Wait, no, let's check again. Wait, Q is at (-1,0), R is at (5,0)? Wait no, looking at the blue figure (QRST), Q is at (-1,0), S is at (3,0)? Wait no, maybe better to check the vertical or horizontal sides. Wait, Q'R' is a side of the smaller figure, length 2 (from x=-1 to x=1, y from 0 to 2? Wait, no, the smaller figure Q'R'S'T' has Q' at (-1,2), R' at (1,2), so Q'R' length is 2 (distance between x=-1 and x=1: 1 - (-1) = 2). The larger figure QRST: Q is at (-1,0), R is at (5,0)? Wait no, S is at (3,-6)? Wait, no, maybe the horizontal side QR: from Q (-1,0) to R (5,0)? No, wait the smaller figure's side Q'R' is length 2, and the larger figure's corresponding side QR: let's see, Q is at (-1,0), R is at (5,0)? Wait, no, maybe the length of Q'R' is 2, and QR is 6? Wait, no, maybe I made a mistake. Wait, the smaller figure (Q'R'S'T'): Q' to R' is 2 units (horizontal), and the larger figure (QRST): Q to R is 6 units? Wait, no, looking at the x-axis: Q is at (-1,0), S is at (1,0) for the smaller? No, no, the smaller figure is above the x-axis, Q' at (-1,2), R' at (1,2), T' at (-2,0), S' at (0,0). So Q'R' is from x=-1 to x=1, so length 2. The larger figure: Q at (-1,0), R at (5,0)? No, S is at (3,-6)? Wait, no, the larger figure's Q is at (-1,0), S is at (3,0)? No, the y-coordinate for the larger figure goes down to -6? Wait, maybe the vertical side: Q' is at (-1,2), T' is at (-2,0), so the vertical distance is 2 (from y=0 to y=2). The larger figure: Q is at (-1,0), T is at (-2,-6), so vertical distance is 6. So scale factor is smaller length / larger length? Wait, no: dilation scale factor is (image length) / (original length). Wait, Q'R'S'T' is the image (after dilation and translation), QRST is the original. So length of Q'R' (image) is 2, length of QR (original) is 6? Wait, no, maybe I got it reversed. Wait, the smaller figure is the image, larger is original. So scale factor = image length / original length. Let's check the horizontal sides: Q'R' (image) length: from x=-1 to x=1, so 2 units. QR (original) length: from x=-1 to x=5? Wait, no, looking at the larger figure, Q is at (-1,0), R is at (5,0)? No, the larger figure's R is at (5,0), S at (3,-6), T at (-2,-6), Q at (-1,0). So QR is from (-1,0) to (5,0), length 6. Q'R' is from (-1,2) to (1,2), length 2. So scale factor is 2/6 = 1/3? Wait, no, that can't be. Wait, maybe the vertical sides: Q' is at (-1,2), T' is at (-2,0), so the vertical segment length is 2 (from y=0 to y=2). Q is at (-1,0), T is at (-2,-6), so vertical segment length is 6. So scale factor is 2/6 = 1/3? Wait, but maybe I mixed up image and original. Wait, 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 scale factor is (length of Q'R') / (length of QR). Let's find coordinates:

For Q'R' (image): Let's take Q'(-1,2) and R'(1,2). The distance between them is $ \sqrt{(1 - (-1))^2 + (2 - 2)^2} = \sqrt{2^2 + 0^2} = 2 $.

For QR (original): Let's take Q(-1,0) and R(5,0). The distance between them is $ \sqrt{(5 - (-1))^2 + (0 - 0)^2} = \sqrt{6^2 + 0^2} = 6 $.

So scale factor = 2 / 6 = 1/3? Wait, no, that seems small. Wait, maybe I got the original and image reversed. Wait, maybe Q'R'S'T' is the original and QRST is the image? No, the problem says "QRST is dilated and translated to form Q'R'S'T'". So QRST is pre-image, Q'R'S'T' is image. So scale factor is image length / pre-image length. So 2 / 6 = 1/3? Wait, but maybe the length of QR is 2 and Q'R' is 6? No, the smaller figure is Q'R'S'T', larger is QRST. Wait, maybe I made a mistake in coordinates. Let's look again:

Smaller 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 x=-1 to x=1, length 2. QRST: Q at (-1,0), R at (5,0), S at (3,-6), T at (-2,-6). So QR is from x=-1 to x=5, length 6. So scale factor is 2/6 = 1/3? Wait, but that would be a reduction. But maybe the vertical side: Q' to T' is from (-1,2) to (-2,0), length $ \sqrt{(-2 - (-1))^2 + (0 - 2)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} $? No, that's not right. Wait, maybe the figures are parallelograms, so we can use horizontal or vertical sides. Wait, Q'R' is horizontal, length 2. QR is horizontal, length 6. So scale factor is 2/6 = 1/3? Wait, but maybe I messed up the original and image. Wait, the problem says "dilated and translated", so dilation is a scale factor. If the image is smaller, scale factor is less than 1. If larger, more than 1. Looking at the graph, Q'R'S'T' is smaller, so scale factor should be less than 1. Wait, but maybe the length of Q'R' is 1 and QR is 3? No, let's count the grid squares. From Q'(-1,2) to R'(1,2): that's 2 grid squares (each grid is 1 unit). From Q(-1,0) to R(5,0): that's 6 grid squares (from x=-1 to x=5 is 6 units). So 2/6 = 1/3. Wait, but maybe the original is Q'R'S'T' and image is QRST? Then scale factor would be 6/2 = 3. But the problem says "QRST is dilated and translated to form Q'R'S'T'". So QRST is pre-image, Q'R'S'T' is image. So scale factor is image length / pre-image length = 2/6 = 1/3. Wait, but that seems incorrect. Wait, maybe I made a mistake in the length of QR. Let's check the x-coordinates: Q is at (-1,0), R is at (3,0)? No, the larger figure's R is at (5,0)? Wait, the x-axis goes from -4 to 4, but R is at x=5? Wait, maybe the grid is 1 unit per square, so from x=-1 to x=1 is 2 units (Q'R'), and from x=-1 to x=3 is 4 units? No, the larger figure's Q is at (-1,0), S is at (3,0), so QS is 4 units? Wait, I'm confused. Let's try another approach. The smaller figure (Q'R'S'T'): the side Q'R' has length 2 (from x=-1 to x=1). The larger figure (QRST): the corresponding side QR has length 6? No, maybe the length of Q'R' is 1 and QR is 3? Wait, no, let's look at the vertical sides. Q' is at (-1,2), T' is at (-2,0): the vertical distance is 2 (from y=0 to y=2). Q is at (-1,0), T is at (-2,-6): vertical distance is 6. So 2/6 = 1/3. So scale factor is 1/3? Wait, but maybe the scale factor is 3? No, because the image is smaller. Wait, maybe I got it reversed. If the original is Q'R'S'T' and image is QRST, then scale factor is 3. But the problem says QRST is dilated to form Q'R'S'T', so QRST is original, Q'R'S'T' is image. So scale factor is image/original = 2/6 = 1/3. Wait, but that seems too small. Wait, maybe the length of Q'R' is 2 and QR is 4? No, let's count the grid squares again. Q' is at (-1,2), R' at (1,2): that's 2 units (x from -1 to 1). Q is at (-1,0), R at (3,0): that's 4 units (x from -1 to 3). Then scale factor is 2/4 = 1/2? No, that's not matching. Wait, maybe the y-coordinate: Q' is at y=2, Q is at y=0. The height of the smaller figure is 2 (from y=0 to y=2), height of larger figure is 6 (from y=0 to y=-6). So 2/6 = 1/3. So scale factor is 1/3.

Wait, maybe I made a mistake. Let's check the coordinates again. Let's list the coordinates:

Smaller figure (Q'R'S'T'):
- Q'(-1, 2)
- R'(1, 2)
- S'(0, 0)
- T'(-2, 0)

Larger figure (QRST):
- Q(-1, 0)
- R(5, 0)
- S(3, -6)
- T(-2, -6)

Now, let's find the length of Q'R' (image) and QR (original):

Q'R' length: distance between (-1,2) and (1,2) is $ \sqrt{(1 - (-1))^2 + (2 - 2)^2} = \sqrt{2^2 + 0^2} = 2 $

QR length: distance between (-1,0) and (5,0) is $ \sqrt{(5 - (-1))^2 + (0 - 0)^2} = \sqrt{6^2 + 0^2} = 6 $

So scale factor = image length / original length = 2 / 6 = 1/3.

Yes, that makes sense. So the scale factor is 1/3.

Answer:

$\frac{1}{3}$