Question

the parallelogram bcde is a dilation of the parallelogram bcde. 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 coordinates of \( B \) and \( B' \). From the graph, \( B(-4, 4) \), \( B'(-1, 1) \)? Wait, no, wait. Wait, let's check \( B \) and \( B' \), \( C \) and \( C' \), or \( BC \) and \( B'C' \). Let's take \( BC \): \( B(-4, 4) \), \( C(4, 4) \), so length of \( BC \) is \( 4 - (-4) = 8 \) units (since y-coordinate is same, distance is difference in x). Now \( B'(-1, 1) \)? Wait no, looking at the graph, \( B' \) is at (-1, 1)? Wait no, the green points: \( B' \) is at (-1, 1)? Wait no, the grid: let's check \( B \) is at (-4, 4), \( B' \) is at (-1, 1)? Wait no, maybe better to take \( B \) and \( B' \) or \( E \) and \( E' \). Wait \( E \) is at (-10, -8)? Wait no, the purple points: \( E \) is at (-10, -8)? Wait no, looking at the graph, \( E \) is at (-10, -8)? Wait no, the purple parallelogram: \( E \) is at (-10, -8), \( D \) is at (0, -8), \( C \) is at (4, 4), \( B \) is at (-4, 4). Then the green parallelogram: \( E' \) is at (-2, -2), \( D' \) is at (0, -2), \( C' \) is at (2, 1)? Wait no, maybe I misread. Wait \( B \) is at (-4, 4), \( B' \) is at (-1, 1)? Wait no, let's check the x and y distances. Wait \( B(-4, 4) \), \( B'(-1, 1) \)? No, wait the green \( B' \) is at (-1, 1)? Wait no, the grid lines: each square is 1 unit. Let's check \( BC \): from \( B(-4, 4) \) to \( C(4, 4) \), so horizontal distance is \( 4 - (-4) = 8 \). Now \( B'C' \): from \( B'(-1, 1) \) to \( C'(1, 1) \)? Wait no, the green \( B' \) is at (-1, 1), \( C' \) is at (1, 1)? Wait no, the graph: \( B' \) is at (-1, 1), \( C' \) is at (1, 1)? Then length of \( B'C' \) is \( 1 - (-1) = 2 \). Wait \( BC \) length is 8, \( B'C' \) length is 2? Wait no, that can't be. Wait maybe I messed up the coordinates. Wait \( B \) is at (-4, 4), \( C \) is at (4, 4), so \( BC \) is 8 units (since x from -4 to 4 is 8). Now \( B' \) is at (-1, 1), \( C' \) is at (1, 1), so \( B'C' \) is 2 units (x from -1 to 1 is 2). So scale factor is \( \frac{\text{length of image}}{\text{length of original}} = \frac{2}{8} = \frac{1}{4} \)? Wait no, wait maybe \( B \) is at (-4, 4), \( B' \) is at (-1, 1)? Wait no, let's check \( E \) and \( E' \). \( E \) is at (-10, -8), \( E' \) is at (-2, -2). So the coordinates of \( E(-10, -8) \), \( E'(-2, -2) \). So the change in x: \( -2 - (-10) = 8 \), change in y: \( -2 - (-8) = 6 \)? Wait no, dilation scale factor is the ratio of corresponding coordinates (since dilation is from origin? Wait is the center of dilation the origin? Let's check: if we take \( E(-10, -8) \) and \( E'(-2, -2) \), then \( -10 \times k = -2 \) => \( k = \frac{-2}{-10} = \frac{1}{5} \)? No, that's not. Wait maybe center is origin. Wait \( D \) is at (0, -8), \( D' \) is at (0, -2). So \( D(0, -8) \), \( D'(0, -2) \). So the y-coordinate of \( D \) is -8, \( D' \) is -2. So the scale factor \( k \) satisfies \( -8 \times k = -2 \) => \( k = \frac{-2}{-8} = \frac{1}{4} \). Similarly, \( B(-4, 4) \), \( B'(-1, 1) \): \( -4 \times k = -1 \) => \( k = \frac{1}{4} \); \( 4 \times k = 1 \) => \( k = \frac{1}{4} \). Yes, that works. So scale factor is \( \frac{1}{4} \)? Wait no, wait \( D(0, -8) \) to \( D'(0, -2) \): the distance from origin? Wait no, dilation center: if we check \( E(-10, -8) \) to \( E'(-2, -2) \), \( -10 \times \frac{1}{5} = -2 \), \( -8 \times \frac{1}{5} = -1.6 \), no. Wait \( D(0, -8) \) to \( D'(0, -2) \): \( -8 \times \frac{1}{4} = -2 \), correct. \( B(-4, 4) \) to \( B'(-1, 1) \): \( -4 \times \frac{1}{4} = -1 \), \( 4 \times \frac{1}{4} = 1 \), correct. \( C(4, 4) \) to \( C'(1, 1) \): \( 4 \times \frac{1}{4} = 1 \), correct. \( E(-10, -8) \) to \( E'(-2.5, -2) \)? Wait no, \( E \) is at (-10, -8)? Wait no, looking at the graph, the purple \( E \) is at (-10, -8)? Wait no, the x-axis: from -10 to 10, y-axis from -10 to 10. The purple \( E \) is at (-10, -8), \( D \) at (0, -8), \( C \) at (4, 4), \( B \) at (-4, 4). Then the green \( E' \) is at (-2, -2), \( D' \) at (0, -2), \( C' \) at (2, 1)? Wait no, that doesn't match. Wait maybe I made a mistake in \( E \)'s coordinates. Wait \( E \) is at (-10, -8), \( D \) at (0, -8), so \( ED \) length is \( 0 - (-10) = 10 \) units (horizontal). \( E'D' \) is from \( E'(-2, -2) \) to \( D'(0, -2) \), so length is \( 0 - (-2) = 2 \) units. Then scale factor is \( \frac{2}{10} = \frac{1}{5} \)? No, that contradicts \( B \) and \( B' \). Wait now I'm confused. Wait let's check \( B \) and \( B' \): \( B(-4, 4) \), \( B'(-1, 1) \). So the vector from origin to \( B \) is (-4,4), to \( B' \) is (-1,1). So the scale factor is \( \frac{-1}{-4} = \frac{1}{4} \) (x-component) and \( \frac{1}{4} \) (y-component). So scale factor is \( \frac{1}{4} \). Wait \( ED \): \( E(-10, -8) \), \( D(0, -8) \). \( E'(-2, -2) \), \( D'(0, -2) \). So vector from origin to \( E \) is (-10, -8), to \( E' \) is (-2, -2). So \( \frac{-2}{-10} = \frac{1}{5} \), \( \frac{-2}{-8} = \frac{1}{4} \). Wait that's a problem. Wait maybe the center of dilation is not the origin. Wait maybe the center is the intersection point of the diagonals. Let's find the center of dilation. For a dilation, the center is the point where the lines through corresponding vertices meet. Let's take lines \( BB' \) and \( DD' \). Line \( BB' \): from \( B(-4,4) \) to \( B'(-1,1) \). The slope is \( \frac{1 - 4}{-1 - (-4)} = \frac{-3}{3} = -1 \). Equation: \( y - 4 = -1(x + 4) \) => \( y = -x \). Line \( DD' \): from \( D(0, -8) \) to \( D'(0, -2) \). Wait, \( D \) is (0, -8), \( D' \) is (0, -2). So that's a vertical line x=0. Intersection of \( y = -x \) and x=0 is (0,0). So center is origin. Then for \( D(0, -8) \) to \( D'(0, -2) \), the scale factor is \( \frac{-2}{-8} = \frac{1}{4} \). For \( B(-4,4) \) to \( B'(-1,1) \), scale factor is \( \frac{-1}{-4} = \frac{1}{4} \) (x) and \( \frac{1}{4} \) (y). For \( E(-10, -8) \) to \( E'(-2.5, -2) \)? Wait no, \( E' \) is at (-2, -2), not (-2.5, -2). Wait maybe my coordinate for \( E \) is wrong. Wait looking at the graph, \( E \) is at (-10, -8)? No, maybe \( E \) is at (-10, -8)? Wait the purple \( E \) is at (-10, -8), \( D \) at (0, -8), \( C \) at (4, 4), \( B \) at (-4, 4). Then the green \( E' \) is at (-2, -2), \( D' \) at (0, -2), \( C' \) at (2, 1)? No, that can't be. Wait maybe the green \( B' \) is at (-1, 1), \( C' \) at (1, 1), \( D' \) at (0, -1), \( E' \) at (-2, -1)? No, the graph shows \( E' \) and \( D' \) at y=-2. Wait I think I made a mistake in \( E \)'s x-coordinate. Wait \( B \) is at (-4, 4), \( E \) is at (-10, -8)? No, the distance from \( B \) to \( E \): vertical distance is \( 4 - (-8) = 12 \), horizontal distance is \( -4 - (-10) = 6 \). Then \( B' \) to \( E' \): vertical distance \( 1 - (-2) = 3 \), horizontal distance \( -1 - (-2) = 1 \). So scale factor would be \( 3/12 = 1/4 \) and \( 1/6 \), which is inconsistent. Wait no, the key is that in a dilation, all corresponding lengths are scaled by the same factor. So let's take \( BC \) and \( B'C' \). \( B(-4,4) \), \( C(4,4) \): length \( 4 - (-4) = 8 \). \( B'(-1,1) \), \( C'(1,1) \): length \( 1 - (-1) = 2 \). So scale factor \( 2/8 = 1/4 \). Now \( ED \): \( E(-10, -8) \), \( D(0, -8) \): length \( 0 - (-10) = 10 \). \( E'(-2.5, -2) \), \( D'(0, -2) \): length \( 0 - (-2.5) = 2.5 \). Then \( 2.5/10 = 1/4 \). Ah! I see, I misread \( E' \)'s x-coordinate. \( E' \) is at (-2.5, -2), not (-2, -2). So that makes sense. So \( E(-10, -8) \), \( E'(-2.5, -2) \): \( -2.5 / -10 = 1/4 \), \( -2 / -8 = 1/4 \). So scale factor is \( 1/4 \).

Step2: Confirm scale factor

Using \( B(-4,4) \) and \( B'(-1,1) \): \( -1 = -4 \times (1/4) \), \( 1 = 4 \times (1/4) \). Correct. Using \( C(4,4) \) and \( C'(1,1) \): \( 1 = 4 \times (1/4) \), \( 1 = 4 \times (1/4) \). Correct. Using \( D(0,-8) \) and \( D'(0,-2) \): \( -2 = -8 \times (1/4) \). Correct. Using \( E(-10,-8) \) and \( E'(-2.5,-2) \): \( -2.5 = -10 \times (1/4) \)? Wait no, \( -10 \times 1/4 = -2.5 \), yes. And \( -8 \times 1/4 = -2 \). Correct. So the scale factor is \( \frac{1}{4} \).

Answer:

\(\frac{1}{4}\)