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: Identify corresponding sides

Find the length of a side in the original figure (QRST) and the corresponding side in the dilated figure (Q'R'S'T'). Let's take the horizontal side (e.g., from Q to S in QRST and Q' to S' in Q'R'S'T').
In QRST, the length from Q (x= -1) to S (x=1) is \(1 - (-1) = 2\)? Wait, no, looking at the grid: Q is at (-1,0), S is at (1,0), so length QS is \(1 - (-1) = 2\)? Wait, no, R is at (4,0), so QR: from Q (-1,0) to R (4,0) is \(4 - (-1) = 5\)? Wait, maybe better to take Q'R' and QR. Q' is at (-1,2), R' is at (1,2), so Q'R' length is \(1 - (-1) = 2\). Q is at (-1,0), R is at (4,0), so QR length is \(4 - (-1) = 5\)? Wait, no, maybe I misread. Wait, the dilated figure (Q'R'S'T') is the smaller one. Wait, original figure QRST: Q is at (-1,0), S is at (1,0)? No, looking at the grid, T' is at (-2,0), Q is at (-1,0), S' is at (1,0), R' is at (2,2)? Wait, maybe better to count grid units. Let's take the vertical side or horizontal side. Let's take the side Q'R': from x=-1 to x=1, so length 2 (since each grid is 1 unit). The corresponding side QR: from x=-1 to x=4? No, wait, original figure QRST: Q is at (-1,0), R is at (4,0), so length QR is \(4 - (-1) = 5\)? No, that can't be. Wait, maybe the original figure's side QS: Q is at (-1,0), S is at (1,0), length 2? No, the dilated figure Q'S' is from x=-1 to x=1, length 2? Wait, no, the dilated figure (Q'R'S'T') has Q' at (-1,2), R' at (1,2), T' at (-2,0), S' at (1,0)? Wait, maybe I made a mistake. Wait, let's look at the y-coordinate. The dilated figure (Q'R'S'T') has height (vertical length) from y=0 to y=2, so height 2. The original figure (QRST) has height from y=0 to y=-6? No, looking at T: T is at (-3,-6), Q is at (-1,0), so the vertical side from Q to T: from y=0 to y=-6, length 6? No, that's not right. Wait, maybe the scale factor is the ratio of the lengths of corresponding sides. Let's take the side Q'R': length is 2 (from x=-1 to x=1, 2 units). The corresponding side QR: from x=-1 to x=4? No, wait, original figure QRST: Q is at (-1,0), R is at (4,0), so length QR is 5? No, that can't be. Wait, maybe the original figure's side QS: Q is at (-1,0), S is at (1,0), length 2. The dilated figure's side Q'S': Q' is at (-1,2), S' is at (1,2)? No, S' is at (1,0)? Wait, the grid: x-axis and y-axis, each grid is 1 unit. Let's re-express coordinates:

Q' : (-1, 2)

R' : (1, 2)

T' : (-2, 0)

S' : (1, 0)

So Q'R' is from (-1,2) to (1,2): length is \(1 - (-1) = 2\) (horizontal distance).

Original figure QRST:

Q : (-1, 0)

R : (4, 0)

T : (-3, -6)

S : (1, -6)

Wait, no, S is at (1,0)? No, looking at the grid, S' is at (1,0), so S is at (1, -6)? No, the original figure is below the x-axis? Wait, Q is at (-1,0), S is at (1,0), T is at (-2,0), R is at (4,0)? No, that's not. Wait, the dilated figure (Q'R'S'T') is above the x-axis, and the original (QRST) is below? No, Q is at (-1,0), so the original figure has Q at (-1,0), T at (-2,0), S at (1,0), R at (4,0), and the dilated figure has Q' at (-1,2), T' at (-2,0), S' at (1,0), R' at (2,2). So the side Q'R' is from (-1,2) to (1,2): length 2 (x from -1 to 1, 2 units). The corresponding side QR: from (-1,0) to (4,0): length 5? No, that's not. Wait, maybe the original figure's side QS: Q at (-1,0), S at (1,0), length 2. The dilated figure's side Q'S': Q' at (-1,2), S' at (1,2)? No, S' is at (1,0). Wait, I think I messed up the coordinates. Let's count the number of grid units between Q' and R': Q' is at x=-1, R' at x=1, so 2 units. Q is at x=-1, R is at x=4? No, R is at x=4? Wait, the original figure's R is at (4,0), so QR is from x=-1 to x=4, length 5. But that can't be. Wait, maybe the scale factor is 1/3? No, wait, let's take the height. The dilated figure (Q'R'S'T') has height from y=0 to y=2, so height 2. The original figure (QRST) has height from y=0 to y=-6? No, T is at (-3,-6), Q is at (-1,0), so the vertical distance is 6 units. Wait, 2/6 = 1/3? No, that's not. Wait, maybe the correct corresponding sides: Q' is at (-1,2), T' is at (-2,0), so the side Q'T' has length. Let's calculate the length of Q'T': distance between (-1,2) and (-2,0): \(\sqrt{(-2 - (-1))^2 + (0 - 2)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}\). Corresponding side QT: Q at (-1,0), T at (-3,-6): distance between (-1,0) and (-3,-6): \(\sqrt{(-3 - (-1))^2 + (-6 - 0)^2} = \sqrt{(-2)^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10}\). No, that's not helpful. Wait, maybe the scale factor is 1/3? No, wait, let's look at the horizontal sides. Q'R' is length 2 (from x=-1 to x=1), QR is length 6 (from x=-1 to x=5? Wait, R is at (4,0), so x=-1 to x=4 is 5 units. No, I think I made a mistake in coordinates. Wait, the dilated figure (Q'R'S'T'): Q' is at (-1,2), R' is at (1,2), so length Q'R' is 2 (1 - (-1) = 2). The original figure (QRST): Q is at (-1,0), R is at (4,0), so length QR is 5 (4 - (-1) = 5). No, that's not. Wait, maybe the original figure's side QS: Q at (-1,0), S at (1,0), length 2. The dilated figure's side Q'S': Q' at (-1,2), S' at (1,2), length 2. No, that would be scale factor 1. But that's not. Wait, maybe the height: Q' is at y=2, Q is at y=0, no. Wait, the dilated figure is above the x-axis, original below? No, Q is at (0,0)? Wait, maybe the grid is 1 unit per square. Let's re-express:

Q' : (-1, 2)

R' : (1, 2)

T' : (-2, 0)

S' : (1, 0)

So Q'R' is horizontal, length 2 (from x=-1 to x=1, 2 units).

Original figure QRST:

Q : (-1, 0)

R : (4, 0)

T : (-3, -6)

S : (1, -6)

Wait, no, S is at (1,0)? No, S' is at (1,0), so S is at (1,0). Then original figure QRST: Q(-1,0), R(4,0), S(1,0), T(-2,0)? No, that's a line. No, the figure is a quadrilateral, so it must have height. Oh, I see, Q'R'S'T' is a parallelogram with height 2 (from y=0 to y=2) and base 2 (from x=-1 to x=1). The original figure QRST is a parallelogram with height 6 (from y=0 to y=-6) and base 6 (from x=-3 to x=3? No, T is at (-3,-6), Q is at (-1,0), S is at (1,0), R is at (3,-6)? Wait, that makes sense. So original figure: T(-3,-6), Q(-1,0), S(1,0), R(3,-6). So base QS: from x=-1 to x=1, length 2? No, Q(-1,0) to S(1,0) is 2 units. R(3,-6) to T(-3,-6) is 6 units. Wait, no, Q(-1,0) to R(3,-6): distance. Wait, the dilated figure Q'R'S'T': Q'(-1,2), R'(1,2), S'(1,0), T'(-2,0). So Q'R' is from (-1,2) to (1,2), length 2. QS is from (-1,0) to (1,0), length 2. No, that can't be. Wait, I think the correct corresponding sides are Q'R' (length 2) and QR (length 4)? No, maybe the scale factor is 1/3. Wait, no, let's count the number of grid units. The dilated figure (Q'R'S'T') has a side length of 2 (from x=-1 to x=1), and the original figure (QRST) has a side length of 6 (from x=-3 to x=3). So scale factor is 2/6 = 1/3? No, that's not. Wait, maybe I made a mistake. Wait, the problem says "dilated and translated", so dilation is a scale factor. Let's take the length of Q'R' and QR. Q'R' is 2 units (from x=-1 to x=1), QR is 4 units? No, Q is at (-1,0), R is at (3,0), so length 4. Then scale factor is 2/4 = 1/2. Wait, that makes sense. Wait, Q' is at (-1,2), R' is at (1,2), so length Q'R' is 2 (1 - (-1) = 2). Q is at (-1,0), R is at (3,0), so length QR is 4 (3 - (-1) = 4). So scale factor is 2/4 = 1/2. Yes, that makes sense. So the scale factor is the ratio of the length of the dilated side to the original side: \( \frac{\text{length of } Q'R'}{\text{length of } QR} = \frac{2}{4} = \frac{1}{2} \). Wait, no, dilated figure is smaller, so scale factor is new length over original length. If original is QR (length 4) and dilated is Q'R' (length 2), then scale factor is 2/4 = 1/2.

Step2: Calculate scale factor

Scale factor \( k = \frac{\text{length of image side}}{\text{length of original side}} \). We found Q'R' (image) length is 2, QR (original) length is 4. So \( k = \frac{2}{4} = \frac{1}{2} \). Wait, but maybe I had the original and image reversed. Wait, dilation: if the image is smaller, scale factor is less than 1. So if QRST is the original, Q'R'S'T' is the image, then scale factor is length of image / length of original. So Q'R' is 2, QR is 4, so 2/4 = 1/2.

Answer:

\(\frac{1}{2}\)