Question

the rectangle cdef is a dilation of the rectangle 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: Find length of original rectangle

First, find the length of side \( CD \) (or \( FE \)) of rectangle \( CDEF \). The coordinates of \( C \) are \((-8, 6)\) and \( D \) are \((-8, -4)\). The distance between them (vertical side) is \( |6 - (-4)| = 10 \)? Wait, no, wait. Wait, actually, let's check the horizontal side. Wait, \( C \) is \((-8,6)\), \( F \) is \((8,6)\). So the horizontal length \( CF \) is \( 8 - (-8) = 16 \)? Wait, no, maybe I messed up. Wait, looking at the grid, the green rectangle: \( C \) is at \((-8,6)\), \( F \) at \((8,6)\), so the length from \( x=-8 \) to \( x=8 \) is \( 16 \)? Wait, no, wait the red rectangle: \( C' \) is at \((-4,3)\)? Wait no, wait the coordinates: \( C' \) is \((-4, 3)\)? Wait no, looking at the graph: \( C' \) is at \((-4, 3)\)? Wait no, the red rectangle \( C'D'E'F' \): \( C' \) is \((-4, 3)\)? Wait no, the y-coordinate of \( C' \) is 3? Wait no, the grid lines: each square is 1 unit. Let's check \( C \): \( x=-8 \), \( y=6 \); \( D \): \( x=-8 \), \( y=-4 \); \( F \): \( x=8 \), \( y=6 \); \( E \): \( x=8 \), \( y=-4 \). So the length of \( CF \) (horizontal side) is \( 8 - (-8) = 16 \)? Wait, no, from \( x=-8 \) to \( x=8 \) is 16 units? Wait, but the red rectangle \( C'D'E'F' \): \( C' \) is at \( x=-4 \), \( F' \) at \( x=4 \). So the horizontal length \( C'F' \) is \( 4 - (-4) = 8 \). Wait, that's 8 units. So original length (CF) is \( 8 - (-8) = 16 \)? Wait, no, wait \( C \) is at \( x=-8 \), \( F \) at \( x=8 \), so the distance is \( 8 - (-8) = 16 \). The red rectangle: \( C' \) at \( x=-4 \), \( F' \) at \( x=4 \), distance is \( 4 - (-4) = 8 \). Wait, but also check the vertical side. \( C \) is at \( y=6 \), \( D \) at \( y=-4 \), so vertical length \( CD \) is \( 6 - (-4) = 10 \)? Wait, no, \( C \) is \((-8,6)\), \( D \) is \((-8,-4)\), so the vertical distance is \( |6 - (-4)| = 10 \). The red rectangle: \( C' \) is \((-4,3)\)? Wait no, looking at the red rectangle: \( C' \) is at \((-4, 3)\)? Wait no, the coordinates: \( C' \) is at \((-4, 3)\)? Wait, no, the y-coordinate of \( C' \) is 3? Wait, no, the graph shows \( C' \) at \((-4, 3)\)? Wait, no, the red rectangle: \( C' \) is at \((-4, 3)\), \( D' \) at \((-4, -1)\)? Wait, no, the red rectangle's \( C' \) is at \((-4, 3)\)? Wait, no, looking at the grid: the red rectangle has \( C' \) at \((-4, 3)\)? Wait, no, the y-axis: the top of red rectangle is at y=3? Wait, no, the original green rectangle: \( C \) is at y=6, \( D \) at y=-4, so the vertical length is \( 6 - (-4) = 10 \). The red rectangle: \( C' \) is at y=3, \( D' \) at y=-1? Wait, no, the red rectangle's \( C' \) is at \((-4, 3)\), \( D' \) at \((-4, -1)\), so vertical length is \( 3 - (-1) = 4 \)? Wait, this is confusing. Wait, maybe I should take the horizontal side. Let's check the horizontal length of the original rectangle (CDEF) between \( C(-8,6) \) and \( F(8,6) \): the distance is \( 8 - (-8) = 16 \) units. The red rectangle (C'D'E'F'): \( C'(-4,3) \) and \( F'(4,3) \): distance is \( 4 - (-4) = 8 \) units. So the scale factor is the ratio of the image length to the original length: \( \frac{8}{16} = \frac{1}{2} \)? Wait, no, wait maybe I got the original and image reversed. Wait, dilation: \( C'D'E'F' \) is a dilation of \( CDEF \), so image is \( C'D'E'F' \), original is \( CDEF \). So scale factor \( k = \frac{\text{length of image}}{\text{length of original}} \). Wait, let's check the horizontal sides. Original \( CF \): from \( x=-8 \) to \( x=8 \), so length \( 16 \). Image \( C'F' \): from \( x=-4 \) to \( x=4 \), length \( 8 \). So \( k = \frac{8}{16} = \frac{1}{2} \). Alternatively, check the vertical sides. Original \( CD \): from \( y=6 \) to \( y=-4 \), length \( 10 \). Image \( C'D' \): from \( y=3 \) to \( y=-1 \), length \( 4 \)? Wait, no, that can't be. Wait, maybe I misread the coordinates. Wait, let's re-express the coordinates:

- \( C \): (-8, 6)
- \( D \): (-8, -4)
- \( F \): (8, 6)
- \( E \): (8, -4)

So the vertical side \( CD \): from \( y=6 \) to \( y=-4 \), so length is \( 6 - (-4) = 10 \).

Now \( C' \): (-4, 3)
\( D' \): (-4, -1)
\( F' \): (4, 3)
\( E' \): (4, -1)

So vertical side \( C'D' \): from \( y=3 \) to \( y=-1 \), length is \( 3 - (-1) = 4 \)? Wait, that's not matching. Wait, maybe I made a mistake in coordinates. Wait, looking at the graph, the red rectangle: \( C' \) is at (-4, 3)? Wait, no, the y-axis: the top of red is at y=3? Wait, the green rectangle's top is at y=6, bottom at y=-4. So the vertical length is 10 (from 6 to -4). The red rectangle's top is at y=3, bottom at y=-1? Wait, no, the red rectangle's \( D' \) is at (-4, -1)? Wait, the graph shows \( D' \) at (-4, -1)? Wait, the original \( D \) is at (-8, -4). Wait, maybe I should check the horizontal side. Original horizontal side: from x=-8 to x=8, so length 16. Red horizontal side: from x=-4 to x=4, length 8. So scale factor is 8/16 = 1/2. Alternatively, vertical side: original vertical side (CD) is from y=6 to y=-4, length 10. Red vertical side (C'D') is from y=3 to y=-1, length 4? Wait, that's not 1/2. Wait, maybe I messed up the coordinates. Wait, let's check \( C' \): the y-coordinate of \( C' \) is 3? Wait, the original \( C \) is at y=6, so 6 to 3 is half. Oh! Wait, 6 * 1/2 = 3. So the y-coordinate of \( C' \) is 3, which is half of 6. Similarly, the x-coordinate of \( C \) is -8, \( C' \) is -4, which is half of -8. So the scale factor is 1/2. Because dilation scale factor is the ratio of the image coordinates to the original coordinates (since dilation is about the origin, probably). Let's confirm: original point \( C(-8,6) \), image \( C'(-4,3) \). So the x-coordinate: -8 * k = -4 ⇒ k = (-4)/(-8) = 1/2. Y-coordinate: 6 * k = 3 ⇒ k = 3/6 = 1/2. So that's consistent. So the scale factor is 1/2.

Step2: Calculate scale factor

The scale factor \( k \) of a dilation is given by the ratio of the length (or coordinate) of the image to the original. For point \( C(-8,6) \) and its image \( C'(-4,3) \), we can calculate \( k \) as \( \frac{-4}{-8} = \frac{1}{2} \) (for x-coordinate) or \( \frac{3}{6} = \frac{1}{2} \) (for y-coordinate). So the scale factor is \( \frac{1}{2} \).

Answer:

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