Calculate Parallelogram Side Length DC Using Expression 5x+12

Q: How do I know which sides are opposite in a parallelogram?

Opposite sides are the ones that don't touch each other. In parallelogram ABCD, sides AB and DC are opposite, while AD and BC are opposite. Think of them as being parallel to each other!

Q: What's the difference between a parallelogram and a rectangle?

A rectangle is a special parallelogram where all angles are 90°. Both have opposite sides equal, but rectangles also have all sides meeting at right angles. The solution method is the same!

Parallelogram Properties with Algebraic Expressions

A parallelogram is shown below.

Calculate the length of the side DC.

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

A parallelogram is shown below.

Calculate the length of the side DC.

Step-by-step solution

To solve this problem, let's begin by identifying the relationship between the sides:

Since $AB = 7x - 4$ and $DC = 5x + 12$ , and we know that opposite sides of a parallelogram are equal, we have:

$7x - 4 = 5x + 12$

First, let's solve for $x$ :

$7x - 5x = 12 + 4$
$2x = 16$

Divide both sides by 2:

$x = 8$

Now that we have the value of $x$ , substitute it back into the expression for $DC$ to find its length:

$DC = 5x + 12 = 5(8) + 12$
$DC = 40 + 12 = 52$

Therefore, the length of side $DC$ is $\mathbf{52}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Property: Opposite sides of a parallelogram are equal in length
Technique: Set expressions equal: $7x - 4 = 5x + 12$
Check: Substitute x = 8 back into DC = 5(8) + 12 = 52 ✓

Common Mistakes

Avoid these frequent errors

Setting adjacent sides equal instead of opposite sides
Don't set AB equal to AD or DC equal to BC = wrong equation! Adjacent sides in a parallelogram are generally not equal. Always identify and set opposite sides equal: AB = DC and AD = BC.

Practice Quiz

Test your knowledge with interactive questions

It is possible to draw a quadrilateral that is not a rectangle, with the sum of its two adjacent angles equaling 180?

FAQ

Everything you need to know about this question

How do I know which sides are opposite in a parallelogram?

Opposite sides are the ones that don't touch each other. In parallelogram ABCD, sides AB and DC are opposite, while AD and BC are opposite. Think of them as being parallel to each other!

Why can't I just substitute any value for x?

The value of x must make the opposite sides equal! Only when $x = 8$ do we get AB = DC = 52. Any other x value would violate the parallelogram property.

What if I get a negative value for x?

Negative x values are mathematically valid! Just make sure that when you substitute back, you get a positive length for the side (since lengths can't be negative in geometry).

Do I need to find the length of all sides?

Not necessarily! The question only asks for DC. Once you solve for x, substitute it into $DC = 5x + 12$ and you're done. But finding AB as a check is always a good idea!

What's the difference between a parallelogram and a rectangle?

A rectangle is a special parallelogram where all angles are 90°. Both have opposite sides equal, but rectangles also have all sides meeting at right angles. The solution method is the same!

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