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In the figure shown, lines AB, CD, and EF are parallel. The length of AC is 2 units, the length of CE is 5 units, the length of AD is (2x + 1) units, and the length of DF is (3x + 6) units. Which is the length of AF?
1. 75 units
2. 5 units
3. 25 units
4. 11 units
To find the length of AF, we first note that since lines AB, CD, and EF are parallel, we can use the properties of similar triangles.
We have:
– AC = 2 units
– CE = 5 units
– AD = (2x + 1) units
– DF = (3x + 6) units
The whole segment AF can be expressed as:
AF = AD + DF
For triangles created by the intersecting lines (which are parallel), we can say that the proportions of the segments formed by the points will be equal.
Since AC (2) relates to CE (5), we can deduce:
AC/CE = AD/DF
Plugging in the values:
2/5 = (2x + 1)/(3x + 6)
Cross-multiplying gives:
2(3x + 6) = 5(2x + 1)
6x + 12 = 10x + 5
4x = 7
x = 7/4
Now, substituting x back into AD and DF:
AD = (2(7/4) + 1) = (14/4 + 4/4) = 18/4 = 9/2 units
DF = (3(7/4) + 6) = (21/4 + 24/4) = 45/4 units
Now, calculate AF:
AF =