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ABCD is a parallelogram with perimeter 40. Find the values of x and y.

ABCD is a parallelogram with perimeter 40. Find the values of x and y.-example-1

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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.

User Wilfried Kopp
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