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 CDEF

Points C(-8,6), D(-8,-4), E(8,-4), F(8,6). Length of CD: vertical distance, \( |6 - (-4)| = 10 \). Length of CF: horizontal distance, \( |8 - (-8)| = 16 \). Wait, no, CDEF is rectangle, so length of CD (vertical side) is \( 6 - (-4) = 10 \)? Wait, no, looking at coordinates: C(-8,6), D(-8,-4): the vertical distance is \( 6 - (-4) = 10 \)? Wait, no, the grid: from y=6 to y=-4 is 10 units? Wait, no, C(-8,6), D(-8,-4): the length is \( |6 - (-4)| = 10 \)? Wait, but C'D'E'F': C'(-4,3), D'(-4,-2), E'(4,-2), F'(4,3). Wait, maybe better to take horizontal side. For CDEF: from x=-8 to x=8, that's 16 units (8 - (-8) = 16). For C'D'E'F': from x=-4 to x=4, that's 8 units (4 - (-4) = 8). So scale factor is new length / original length = 8 / 16 = 1/2. Wait, let's check vertical side. CDEF: y from 6 to -4, length 10? No, wait C(-8,6), D(-8,-4): the vertical length is 6 - (-4) = 10? But C'(-4,3), D'(-4,-2): vertical length is 3 - (-2) = 5. So 5 / 10 = 1/2. Yes, that matches. So scale factor is 1/2.

Step2: Calculate scale factor

Scale factor = length of C'D'E'F' / length of CDEF. Take horizontal side: CDEF horizontal length: 8 - (-8) = 16. C'D'E'F' horizontal length: 4 - (-4) = 8. So scale factor = 8 / 16 = 1/2.

Answer:

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