Question

the parallelogram jklm is a dilation of the parallelogram jklm. 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: Identify coordinates of corresponding points

First, find the coordinates of a point from the original parallelogram (JKLM) and its image (J'K'L'M'). Let's take point K and K'. From the graph, K is at (2, 0) and K' is at (10, 0).

Step2: Calculate the scale factor

The scale factor \( k \) of a dilation is given by the ratio of the distance from the center to the image point to the distance from the center to the original point. Since the center of dilation seems to be the origin (as the lines from the origin pass through corresponding points), we can also use the ratio of the coordinates. For the x - coordinate (since y - coordinate is 0 for K and K'), the scale factor \( k=\frac{\text{Coordinate of }K'}{\text{Coordinate of }K}=\frac{10}{2} = 5\)? Wait, no, wait. Wait, let's check another point. Let's take point J: J is at (1, 0) and J' is at (5, 0). Wait, maybe I misread the coordinates. Wait, looking at the graph, J is at (1, 0)? Wait, no, the grid: J is at (1, 0)? Wait, no, the blue points: J is at (1, 0)? Wait, no, the x - axis: the blue points J and K are at (1, 0) and (2, 0)? Wait, no, the grid lines: each grid is 1 unit. Wait, M is at (-3, -2), L is at (-2, -2), J is at (1, 0), K is at (2, 0). Then M' is at (-10, -10)? Wait, no, M' is at (-10, -10)? Wait, no, the purple points: M' is at (-10, -10), L' is at (-6, -10)? Wait, no, maybe I made a mistake. Wait, let's take the length of JK and J'K'. The length of JK: distance between J(1, 0) and K(2, 0) is \( 2 - 1=1 \)? Wait, no, wait the x - coordinates: J is at (1, 0)? Wait, no, looking at the graph, the blue points: J is at (1, 0)? Wait, no, the grid: the first blue point (J) is at (1, 0), K at (2, 0), L at (-2, -2), M at (-3, -2). Then the purple points: J' at (5, 0), K' at (10, 0), L' at (-6, -10)? Wait, no, maybe the center is not the origin. Wait, let's calculate the vector or the ratio of the lengths. The length of JK: from x = 1 to x = 2, so length 1? No, wait, maybe J is at (1, 0) and J' is at (5, 0), so the ratio is 5/1 = 5? No, that can't be. Wait, wait, maybe I misread the coordinates. Wait, let's look again. The original parallelogram JKLM: points J, K, L, M. J is at (1, 0), K at (2, 0), L at (-2, -2), M at (-3, -2). The dilated parallelogram J'K'L'M': J' at (5, 0), K' at (10, 0), L' at (-6, -10), M' at (-10, -10). Wait, no, the distance from J to the origin: J is at (1, 0), J' is at (5, 0). So the scale factor is 5? Wait, no, wait the vector from J to K is (1, 0), and from J' to K' is (5, 0). So the scale factor is 5? Wait, but let's check the y - component. The original height: from y = 0 to y = -2, so height 2. The dilated height: from y = 0 to y = -10, so height 10. So 10/2 = 5. So the scale factor is 5? Wait, no, wait maybe I made a mistake in coordinates. Wait, let's take point M: M is at (-3, -2), M' is at (-10, -10)? No, that doesn't fit. Wait, maybe the original points: J is at (1, 0), K at (2, 0), L at (-1, -2), M at (-2, -2). Ah! That makes more sense. So J(1, 0), K(2, 0), L(-1, -2), M(-2, -2). Then J'(5, 0), K'(10, 0), L'(-5, -10), M'(-10, -10). Now, the distance from J to K: \( 2 - 1 = 1 \) (x - distance), and from J' to K': \( 10 - 5 = 5 \). The y - distance from J (y=0) to L (y = - 2) is 2, and from J' (y=0) to L' (y=-10) is 10. So the scale factor is \( \frac{5}{1}=5 \) or \( \frac{10}{2}=5 \)? Wait, no, J is at (1, 0), J' is at (5, 0), so the scale factor is \( \frac{5}{1}=5 \)? Wait, no, 5/1 is 5? But let's check the x - coordinate of J: if J is at (1, 0), J' is at (5, 0), then the scale factor is 5. But wait, maybe the original J is at (1, 0) and J' is at (5, 0), so 5/1 = 5. But wait, let's check the vector from the center. Wait, maybe the center is the origin, and the scale factor is 5? Wait, no, that seems too big. Wait, maybe I misread the coordinates. Wait, let's look at the length of JK: JK is from (1, 0) to (2, 0), length 1. J'K' is from (5, 0) to (10, 0), length 5. So the scale factor is 5? Wait, but that seems large. Wait, maybe the original points are J(1, 0), K(2, 0), so the vector JK is (1, 0). The vector J'K' is (5, 0), so the scale factor is 5. Alternatively, take point M(-3, -2) and M'(-15, -10)? No, the purple M' is at (-10, -10). Wait, -3 times what is -10? No, that's not. Wait, maybe the center is not the origin. Wait, let's find the center of dilation. The center of dilation is the intersection point of the lines joining corresponding vertices. So line MM' and line KK' and line JJ' and line LL' should meet at the center. Let's find the equation of line KK': K(2, 0) and K'(10, 0) is a horizontal line y = 0. Line MM': M(-3, -2) and M'(-10, -10). The slope of MM' is \( \frac{-10 - (-2)}{-10 - (-3)}=\frac{-8}{-7}=\frac{8}{7} \)? No, that can't be. Wait, maybe I have the coordinates wrong. Let's look at the graph again. The blue parallelogram: J is at (1, 0), K at (2, 0), L at (-2, -2), M at (-3, -2). The purple parallelogram: J' at (5, 0), K' at (10, 0), L' at (-6, -10), M' at (-10, -10). Wait, the vector from J to J' is (4, 0)? No, 5 - 1 = 4. From K to K' is 10 - 2 = 8. Wait, that's not a scale factor. Wait, maybe the original J is at (1, 0) and J' is at (5, 0), so 5/1 = 5. K is at (2, 0), K' at (10, 0), 10/2 = 5. L is at (-1, -2), L' at (-5, -10), -5/-1 = 5, -10/-2 = 5. Ah! There we go. So L is at (-1, -2), L' is at (-5, -10). So the x - coordinate: -5 / -1 = 5, y - coordinate: -10 / -2 = 5. So the scale factor is 5? Wait, no, -1 times 5 is -5, -2 times 5 is -10. Yes. So the scale factor is 5? Wait, but that seems large. Wait, maybe I misread L's coordinate. Let's check L: in the blue parallelogram, L is at (-2, -2)? No, the blue points: M is at (-3, -2), L is at (-2, -2), J is at (1, 0), K is at (2, 0). Then L' is at (-6, -10)? No, -2 times 5 is -10? No, -2 times 5 is -10, but -6 is not -2 times 3. Wait, I'm confused. Wait, let's calculate the distance between J and K: J(1, 0), K(2, 0), distance is 1. Distance between J'(5, 0) and K'(10, 0) is 5. So 5/1 = 5. Distance between M(-3, -2) and L(-2, -2) is 1. Distance between M'(-10, -10) and L'(-6, -10)? No, that's 4. Wait, no, I must have the coordinates wrong. Wait, the purple parallelogram: M' is at (-10, -10), L' is at (-6, -10), J' at (5, 0), K' at (10, 0). So the length of L'M' is 10 - 6 = 4? No, -6 to -10 is 4 units? No, -6 - (-10)=4. The length of ML is -2 - (-3)=1. So 4/1 = 4? No, that's not. Wait, maybe the center is not the origin. Let's find the center of dilation. The center (h, k) satisfies the equation that for any point (x, y) and its image (x', y'), \( x'=h + k(x - h) \), \( y'=k + k(y - k) \), where k is the scale factor. Let's take point J(1, 0) and J'(5, 0). So \( 5=h + k(1 - h) \), \( 0=k + k(0 - k) \). From the y - equation: \( 0 = k - k^2 \), \( k(1 - k)=0 \), so k = 0 or k = 1. But k = 0 is not possible, k = 1 is not possible. So my coordinate reading is wrong. Wait, maybe J is at (1, 0) and J' is at (5, 0), but the center is not the origin. Wait, let's look at the graph again. The blue parallelogram: J is at (1, 0), K at (2, 0), L at (-1, -2), M at (-2, -2). The purple parallelogram: J' at (5, 0), K' at (10, 0), L' at (-5, -10), M' at (-10, -10). Now, the vector from J to K is (1, 0), from J' to K' is (5, 0). The vector from M to L is (1, 0), from M' to L' is (5, 0). The vector from J to M is (-2, -2), from J' to M' is (-10, -10). Ah! So the vector J'M' is 5 times the vector JM: (-10, -10)=5*(-2, -2). So the scale factor is 5? Wait, -2*5=-10, -2*5=-10. Yes! So the vector from J to M is (-2, -2), and from J' to M' is (-10, -10), which is 5 times (-2, -2). So the scale factor is 5? Wait, but that seems large. Wait, maybe the original J is at (1, 0) and J' is at (5, 0), so the scale factor is 5. But let's check the length of JM: \( \sqrt{(-2)^2+(-2)^2}=\sqrt{8}=2\sqrt{2} \). The length of J'M': \( \sqrt{(-10)^2+(-10)^2}=\sqrt{200}=10\sqrt{2} \). Then \( \frac{10\sqrt{2}}{2\sqrt{2}} = 5 \). Yes, so the scale factor is 5. Wait, but I think I made a mistake earlier in the coordinate of L. L should be (-1, -2) if M is (-2, -2), because ML is a horizontal line (same y - coordinate) with length 1. Then L' is (-5, -10), which is 5 times (-1, -2). Yes, that works. So the scale factor is 5.

Step3: Confirm with another point

Take point K(2, 0) and K'(10, 0). The scale factor is \( \frac{10}{2}=5 \). Take point L(-1, -2) and L'(-5, -10). The scale factor is \( \frac{-5}{-1}=5 \) (for x - coordinate) and \( \frac{-10}{-2}=5 \) (for y - coordinate). So the scale factor is 5.

Answer:

\( 5 \)