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If ON = 8x − 8, LM = 7x + 4, NM = x − 5, and OL = 3y − 6, find the values of x and y for which LMNO must be a parallelogram.
For a quadrilateral to be a parallelogram, opposite sides must be equal. Therefore, we can set up the following equations:
ON = LM
8x - 8 = 7x + 4
Solving this equation for x:
8x - 7x = 4 + 8
x = 12
Now let's check the other pair of opposite sides:
NM = OL
x - 5 = 3y - 6
Solving this equation for y:
3y - x = 6 - 5
3y - x = 1
Since we have two variables in this equation, we can't find a unique solution. However, if we substitute x = 12 into the equation, we can find the corresponding value of y:
3y - 12 = 1
3y = 13
y = 13/3
Therefore, for LMNO to be a parallelogram, the values of x and y must be x = 12 and y = 13/3.