If ON=8x-8, LM=7x+4, NM=x-5, and OL=3y-8, find the values of x and y for which LMNO must be a parallelogram.
In order for LMNO to be a parallelogram, opposite sides must be congruent. Therefore, we can set up two equations using the given side lengths:
OL = NM
3y - 8 = x - 5
ON = LM
8x - 8 = 7x + 4
Solving the first equation for x, we get:
x = 3y - 3
Substituting this expression for x into the second equation, we have:
8(3y - 3) - 8 = 7(3y - 3) + 4
24y - 24 - 8 = 21y - 21 + 4
24y - 32 = 21y - 17
3y = 15
y = 5
Substituting this value of y into the expression for x:
x = 3(5) - 3
x = 12
Therefore, x = 12 and y = 5.