Since ABCD is a parallelogram, opposite sides are equal in length. Let's denote the length of side AB as x and the length of side BC as y. Then, we have:
AB = CD = x
BC = AD = y
The perimeter of the parallelogram is given as 40, so we have:
AB + BC + CD + AD = 40
Substituting the values of AB, BC, CD, and AD, we get:
x + y + x + y = 40
Simplifying this equation, we get:
2x + 2y = 40
Dividing both sides by 2, we get:
x + y = 20
We have one equation with two unknowns, x and y. However, we know that opposite sides of a parallelogram are parallel and equal in length. Therefore, we can use this information to find another equation involving x and y. Since AB is parallel to CD and BC is parallel to AD, we have:
ABCD is a parallelogram
=> AB || CD and BC || AD
Thus, we have:
BC = AD = y
and
AB = CD = x
Now, we can use the fact that opposite sides of a parallelogram are equal in length to get another equation:
Perimeter of ABCD = 2(AB + BC)
=> 40 = 2(x + y)
Simplifying this equation, we get:
x + y = 20
We can see that this is the same equation we obtained earlier. Therefore, we have two equations with two unknowns:
x + y = 20
2x + 2y = 40
We can solve this system of equations by either substitution or elimination. Let's use the substitution method:
x + y = 20 (equation 1)
2x + 2y = 40 (equation 2)
From equation 1, we can solve for y in terms of x:
y = 20 - x
Substituting this expression for y into equation 2, we get:
2x + 2(20 - x) = 40
Simplifying and solving for x, we get:
2x + 40 - 2x = 40
=> 40 = 40
This equation is true for any value of x, which means that x can take on any value. Substituting this value for y in equation 1, we get:
x + y = 20
x + (20 - x) = 20
=> y = 20 - x
Therefore, the values of x and y can be any pair of numbers that satisfy the equation x + y = 20. For example, x = 5 and y = 15, or x = 8 and y = 12, or x = 10 and y = 10.