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 \( CF \) (or \( DE \)) of rectangle \( CDEF \). Coordinates of \( C(-8,6) \) and \( F(8,6) \). The length is \( 8 - (-8)=16 \)? Wait, no, wait. Wait, looking at the graph, \( C \) is at \( (-8,6) \), \( F \) is at \( (8,6) \)? Wait no, wait the x-coordinate of \( C \) is -8, \( F \) is 8? Wait no, the grid: let's check the x-coordinates. Wait \( C \) is at \( (-8,6) \), \( F \) is at \( (8,6) \)? Wait no, the horizontal distance between \( C(-8,6) \) and \( F(8,6) \) is \( 8 - (-8)=16 \)? Wait but the red rectangle \( C'D'E'F' \): \( C'(-4,3) \)? Wait no, \( C' \) is at \( (-4,3) \)? Wait no, the coordinates: \( C' \) is at \( (-4,3) \)? Wait the y-coordinate of \( C' \) is 3? Wait no, the graph: \( C' \) is at \( (-4,3) \)? Wait no, looking at the y-axis: \( C \) is at y=6, \( C' \) is at y=3? Wait no, the red rectangle: \( C'(-4,3) \), \( F'(4,3) \). So the length of \( C'F' \) is \( 4 - (-4)=8 \). The length of \( CF \) is \( 8 - (-8)=16 \)? Wait no, wait \( C \) is at \( (-8,6) \), \( F \) is at \( (8,6) \), so the horizontal distance is \( 8 - (-8)=16 \). \( C' \) is at \( (-4,3) \), \( F' \) is at \( (4,3) \), horizontal distance is \( 4 - (-4)=8 \). So the scale factor is \( \frac{\text{length of image}}{\text{length of original}}=\frac{8}{16}=\frac{1}{2} \)? Wait or check the vertical side. \( C(-8,6) \), \( D(-8,-4) \), so vertical length is \( 6 - (-4)=10 \)? Wait no, \( D \) is at \( (-8,-4) \), so vertical distance from \( C(-8,6) \) to \( D(-8,-4) \) is \( 6 - (-4)=10 \)? Wait \( C'(-4,3) \), \( D'(-4,-1) \), vertical distance is \( 3 - (-1)=4 \)? Wait that can't be. Wait maybe I misread the coordinates. Let's re-examine:

Wait \( C \) is at \( (-8,6) \), \( D \) is at \( (-8,-4) \), so the vertical side \( CD \) has length \( 6 - (-4)=10 \)? Wait no, \( y \)-coordinate of \( C \) is 6, \( D \) is -4, so difference is \( 6 - (-4)=10 \). \( C' \) is at \( (-4,3) \), \( D' \) is at \( (-4,-1) \), so vertical length \( C'D' \) is \( 3 - (-1)=4 \)? That doesn't match. Wait maybe I got the coordinates wrong. Let's look at the graph again. The green rectangle: \( C \) is at \( (-8,6) \), \( D \) is at \( (-8,-4) \), \( E \) is at \( (8,-4) \), \( F \) is at \( (8,6) \). So the length of \( CD \) (vertical side) is \( 6 - (-4)=10 \), and length of \( CF \) (horizontal side) is \( 8 - (-8)=16 \). Now the red rectangle: \( C'(-4,3) \), \( D'(-4,-1) \), \( E'(4,-1) \), \( F'(4,3) \). So length of \( C'D' \) (vertical side) is \( 3 - (-1)=4 \)? Wait no, \( 3 - (-1)=4 \), but original \( CD \) is 10? That can't be. Wait maybe the y-coordinate of \( C \) is 6, \( C' \) is 3, so the vertical distance from \( C \) to \( D \) is \( 6 - (-4)=10 \), and from \( C' \) to \( D' \) is \( 3 - (-1)=4 \)? No, that's not a dilation. Wait maybe I made a mistake in coordinates. Wait the grid lines: each grid is 1 unit. Let's check \( C \): x=-8, y=6. \( C' \): x=-4, y=3. So the x-coordinate of \( C \) is -8, \( C' \) is -4: so the horizontal distance from the origin? Wait dilation center: probably the origin (0,0), since the coordinates are scaled. So the scale factor is the ratio of the coordinates of the image to the original. For point \( C(-8,6) \), the image \( C'(-4,3) \). So the x-coordinate of \( C' \) is \( -4 = -8 \times \frac{1}{2} \), y-coordinate of \( C' \) is \( 3 = 6 \times \frac{1}{2} \). Ah! So that's the key. So the coordinates of \( C \) are \( (-8,6) \), \( C' \) are \( (-4,3) \). So the scale factor is \( \frac{-4}{-8}=\frac{1}{2} \) (for x-coordinate) and \( \frac{3}{6}=\frac{1}{2} \) (for y-coordinate). So the scale factor is \( \frac{1}{2} \).

Step2: Verify with another point

Take point \( D(-8,-4) \), image \( D'(-4,-2) \). So x-coordinate: \( -4 = -8 \times \frac{1}{2} \), y-coordinate: \( -2 = -4 \times \frac{1}{2} \). Perfect, that matches. So the scale factor is \( \frac{1}{2} \).

Answer:

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