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Answer :
Final answer:
To find the length of DF, given the ratio of lengths CD:DE:EF as 3:2:1 and CF = 42, we first identify the lengths as multiples of a variable x, solve for x, and then calculate DF, resulting in a length of 21.
Explanation:
The question asks to find the length of DF in a line segment, given that points C, D, E, and F lie in that order on the same line segment and the lengths CD, DE, and EF have the ratio 3:2:1. Furthermore, it's given that CF = 42. First, we can denote the lengths of CD, DE, and EF as 3x, 2x, and x, respectively, since their ratio is 3:2:1. Hence, the total length of CF (which is CD + DE + EF) is 3x + 2x + x = 6x. Given that CF = 42, we can solve for x by setting 6x = 42, which gives x = 7.
To find DF, we sum the lengths DE and EF, which are 2x and x, respectively. Thus, DF = 2x + x = 3x. Substituting x = 7 into this equation gives DF = 3(7) = 21. Therefore, the length of DF is 21.
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