Question

the trapezoid cdef is a dilation of the trapezoid cdef. 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 trapezoid \( CDEF \) and its image \( C'D'E'F' \). Let's take point \( D \) and \( D' \). From the graph, \( D \) has coordinates \( (-2, 2) \) and \( D' \) has coordinates \( (-3, 3) \)? Wait, no, let's check again. Wait, maybe better to take \( E \) and \( E' \). \( E \) is at \( (-2, 0) \), \( E' \) is at \( (-3, 0) \)? Wait, no, looking at the grid: Let's take point \( C \): \( C \) is at \( (2, 2) \), \( C' \) is at \( (3, 3) \)? Wait, no, maybe I misread. Wait, original trapezoid \( CDEF \): Let's find coordinates of \( D \): \( D \) is at \( (-2, 2) \), \( D' \) is at \( (-3, 3) \)? Wait, no, the blue points are \( C'D'E'F' \), pink are \( CDEF \). So \( D \) (pink) is at \( (-2, 2) \), \( D' \) (blue) is at \( (-3, 3) \)? Wait, no, let's check the x and y distances. Wait, maybe \( D \) is at \( (-2, 2) \), \( D' \) is at \( (-3, 3) \)? Wait, no, the grid lines: each square is 1 unit. Let's take point \( E \): \( E \) (pink) is at \( (-2, 0) \), \( E' \) (blue) is at \( (-3, 0) \)? Wait, no, \( E' \) is at \( (-3, 0) \)? Wait, the x-coordinate of \( E \) is -2, \( E' \) is -3? No, wait, maybe \( D \) (pink) is at \( (-2, 2) \), \( D' \) (blue) is at \( (-3, 3) \)? Wait, no, let's calculate the distance from the origin or the scale factor as the ratio of the length of the image segment to the original segment. Let's take the vertical side from \( D \) to \( E \): \( D \) is at \( (-2, 2) \), \( E \) is at \( (-2, 0) \), so the length is \( 2 - 0 = 2 \). For \( D' \) to \( E' \): \( D' \) is at \( (-3, 3) \), \( E' \) is at \( (-3, 0) \), so the length is \( 3 - 0 = 3 \)? Wait, no, that can't be. Wait, maybe I got the points wrong. Wait, original trapezoid \( CDEF \): Let's check \( C \): \( C \) (pink) is at \( (2, 2) \), \( C' \) (blue) is at \( (3, 3) \)? Wait, no, the pink points: \( C \) is at \( (2, 2) \), \( D \) at \( (-2, 2) \), \( E \) at \( (-2, 0) \), \( F \) at \( (2, -2) \). Then \( C' \) is at \( (3, 3) \), \( D' \) at \( (-3, 3) \), \( E' \) at \( (-3, 0) \), \( F' \) at \( (3, -3) \). Ah, there we go. So \( D \) is \( (-2, 2) \), \( D' \) is \( (-3, 3) \). Wait, no, \( D \) (pink) is \( (-2, 2) \), \( D' \) (blue) is \( (-3, 3) \)? Wait, no, the x-coordinate of \( D \) is -2, \( D' \) is -3? Wait, no, the distance between \( D \) and \( D' \): the x-coordinate of \( D \) is -2, \( D' \) is -3? No, that's a shift. Wait, no, dilation is a scale factor, so the ratio of the length of \( D'E' \) to \( DE \). \( DE \): from \( (-2, 2) \) to \( (-2, 0) \), length is \( 2 - 0 = 2 \) (vertical segment). \( D'E' \): from \( (-3, 3) \) to \( (-3, 0) \), length is \( 3 - 0 = 3 \)? Wait, no, that would be scale factor \( 3/2 \), but that doesn't seem right. Wait, maybe I mixed up the points. Wait, let's take \( E \) (pink) at \( (-2, 0) \), \( E' \) (blue) at \( (-3, 0) \)? No, \( E' \) is at \( (-3, 0) \)? Wait, the grid: each square is 1 unit. Let's check the x-coordinate of \( E \): \( E \) is at \( x = -2 \), \( E' \) is at \( x = -3 \)? No, that's a horizontal shift. Wait, maybe the center of dilation is the origin? Let's check the coordinates of \( C \) (pink) at \( (2, 2) \), \( C' \) (blue) at \( (3, 3) \). The vector from origin to \( C \) is \( (2, 2) \), to \( C' \) is \( (3, 3) \). So the scale factor is \( 3/2 \)? Wait, no, \( 3/2 \) is 1.5, but let's check another point. \( F \) (pink) is at \( (2, -2) \), \( F' \) (blue) is at \( (3, -3) \). So the vector from origin to \( F \) is \( (2, -2) \), to \( F' \) is \( (3, -3) \). So the scale factor is \( 3/2 \)? Wait, no, that can't be. Wait, maybe the original trapezoid \( CDEF \) has \( D \) at \( (-2, 2) \), \( C \) at \( (2, 2) \), \( E \) at \( (-2, 0) \), \( F \) at \( (2, -2) \). Then \( D' \) is at \( (-3, 3) \), \( C' \) at \( (3, 3) \), \( E' \) at \( (-3, 0) \), \( F' \) at \( (3, -3) \). So the length of \( CD \) (original) is \( 2 - (-2) = 4 \) (horizontal segment from \( x=-2 \) to \( x=2 \), y=2). The length of \( C'D' \) (image) is \( 3 - (-3) = 6 \) (horizontal segment from \( x=-3 \) to \( x=3 \), y=3). So scale factor is \( 6/4 = 3/2 \)? Wait, no, that's 1.5. But let's check the vertical segment \( DE \): \( D \) at \( (-2, 2) \), \( E \) at \( (-2, 0) \), length 2. \( D'E' \): \( D' \) at \( (-3, 3) \), \( E' \) at \( (-3, 0) \), length 3. So \( 3/2 \) is the scale factor. Wait, but maybe I made a mistake. Wait, the problem says "the trapezoid \( C'D'E'F' \) is a dilation of \( CDEF \)". So the scale factor is the ratio of the length of a side in \( C'D'E'F' \) to the corresponding side in \( CDEF \). Let's take \( DE \): length is 2 (from y=2 to y=0, x=-2). \( D'E' \): length is 3 (from y=3 to y=0, x=-3). So scale factor is \( 3/2 \)? Wait, no, \( 3/2 \) is 1.5, but let's check the coordinates again. Wait, maybe \( D \) is at \( (-2, 2) \), \( D' \) is at \( (-3, 3) \): the x-coordinate changes from -2 to -3 (difference of -1), y from 2 to 3 (difference of +1). But dilation is about scaling, not shifting. Wait, maybe the center of dilation is the origin, so the coordinates of \( D' \) are \( k \times \) coordinates of \( D \), where \( k \) is the scale factor. So \( D \) is \( (-2, 2) \), \( D' \) is \( (-3, 3) \). So \( -3 = k \times (-2) \), so \( k = 3/2 \). Similarly, \( 3 = k \times 2 \), so \( k = 3/2 \). Yes, that works. So the scale factor is \( 3/2 \)? Wait, but let's check \( E \): \( E \) is \( (-2, 0) \), \( E' \) is \( (-3, 0) \). So \( -3 = k \times (-2) \), so \( k = 3/2 \), and \( 0 = k \times 0 \), which holds. So the scale factor is \( 3/2 \). Wait, but maybe I misread the coordinates. Wait, the pink point \( D \) is at \( (-2, 2) \), blue \( D' \) at \( (-3, 3) \). So the x-coordinate of \( D \) is -2, \( D' \) is -3: \( -3 = (3/2) \times (-2) \), yes, because \( (3/2) \times (-2) = -3 \). Similarly, y-coordinate: \( (3/2) \times 2 = 3 \), which matches \( D' \)'s y-coordinate. So the scale factor is \( 3/2 \). Wait, but maybe the original trapezoid has \( D \) at \( (-2, 2) \), and \( D' \) at \( (-3, 3) \), so the scale factor is \( 3/2 \).

Step2: Calculate the scale factor

The scale factor \( k \) of a dilation is the ratio of the length of a side in the image to the length of the corresponding side in the original figure. Using the coordinates of a point \( (x, y) \) in the original figure and \( (kx, ky) \) in the image (if the center of dilation is the origin), we can find \( k \). For point \( D(-2, 2) \) and \( D'(-3, 3) \), we have \( -3 = k \times (-2) \) and \( 3 = k \times 2 \). Solving for \( k \) from \( 3 = k \times 2 \), we get \( k = \frac{3}{2} \). We can verify with another point, say \( F(2, -2) \) and \( F'(3, -3) \): \( 3 = k \times 2 \) gives \( k = \frac{3}{2} \), and \( -3 = k \times (-2) \) also gives \( k = \frac{3}{2} \). Thus, the scale factor is \( \frac{3}{2} \).

Answer:

\(\frac{3}{2}\)