13. The midpoint divides the segment into two equal parts. That is
BC = (1/2)·AC = (1/2)·(8x -20)
BC = 4x -10
Since you are told the other half is 3x -1, the two halves being equal tells you
... AB = BC
... 3x -1 = 4x -10
... x = 9 . . . . . add 10-3x
Then the measures of AB and BC are 3·9 - 1 = 4·9 -10 = 26.
BC = 26
The problem statement is not clear as to the desired form of the answer—whether it is to be in terms of x, or a number.
14. Same deal as the previous problem. The halves of the bisected segment are equal, giving you an equation that can be solved for x.
... CG = GD
... 5x -1 = 7x -13
... 12 = 2x . . . . . . . . . add 13-5x
... x = 6 . . . . . . . . . . divide by 2
Now you have a value of x that you can use to find the length of EF.
... EF = 6x - 4 = 6·6 - 4 = 32
The parts of EF add to give its total length, so
... EG + GF = EF
... EG + 13 = 32
... EG = 19
15. Once again, the parts of a line segment add to give the whole.
... RS + ST = RT
... (2x -4) + (4x -1) = (8x -43)
... 38 = 2x . . . . . . . . . . . . . . . . add 43-6x
... x = 19
Since R is the midpoint of QS, RS will be half the length of QS. The length of RS is 2x -4 = 2(19) - 4 = 34. Then the length of QS is 2·34 = 68.
QS = 68
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To help check our work, we can find the lengths of the other segments in the diagram. ST = 4x - 1 = 75; RT = 8x - 43 = 109. So, we have ...
... QR : RS : ST = 34 : 34 : 75 . . . with QS = 68 and RT = 109.
