Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.

Question

The figure is quadrilateral L M N O. It appears to be a parallelogram.

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. The diagram is not drawn to scale.

Answer 1

To determine the values of x and y for which quadrilateral LMNO must be a parallelogram, we need to use the properties of parallelograms.

One property of parallelograms is that opposite sides are congruent. In quadrilateral LMNO, the opposite sides are LM and NO.

Given: LM = 7x + 4 and NO = ON = 8x - 8

To make LMNO a parallelogram, we set LM equal to NO:

7x + 4 = 8x - 8

Now, we can solve this equation for x:

7x - 8x = -8 - 4
-x = -12
x = 12

Now that we have determined x, we can find the value of y using the given information:

OL = 3y - 6

Since opposite sides in a parallelogram are congruent, OL must also be equal to NM:

3y - 6 = x - 5

Substitute x = 12 into the equation:

3y - 6 = 12 - 5
3y - 6 = 7
3y = 7 + 6
3y = 13
y = 13/3

Therefore, LMNO must be a parallelogram when x = 12 and y = 13/3.