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Since LL is the midpoint of start overline, I, K, end overline IK, we know that LL = 1/2(IK). Similarly, MM is the midpoint of start overline, J, K, end overline JK, so MM = 1/2(JK).
We are given that LM = 8x - 53 and IJ = -26 + 6x.
Since LL = 1/2(IK), we have LL = 1/2(IK) = 1/2(IJ + JK) = 1/2(IJ + MM), then
8x - 53 = 1/2(-26 + 6x + MM).
We know that MM = 1/2(JK), so MM = 1/2(IJ + JK). Since IJ = -26 + 6x, then JK = MM = 1/2(IJ) = 1/2(-26 + 6x) = -13 + 3x.
Substitute this into the equation above:
8x - 53 = 1/2(-26 + 6x + (-13 + 3x))
8x - 53 = 1/2(-39 + 9x)
16x - 106 = -39 + 9x
7x = 67
x = 67/7 = 9.57
So, the measure of LM is LM = 8(9.57) - 53 = 20.56.