Solve for X: Parallelogram with Perimeter 30 and Side Length 8

Q: Why do we only need two side lengths for a parallelogram's perimeter?

In a parallelogram, opposite sides are always equal. So if AB = 8, then the opposite side CD also equals 8. If AC = X+2, then the opposite side BD also equals X+2.

Q: Can I solve this without the perimeter formula?

Yes, but it's harder! You could write out all four sides: $ 8 + (X+2) + 8 + (X+2) = 30 $. The formula $ P = 2(a + b) $ just makes it cleaner.

Q: How do I verify my answer makes sense?

Two checks: (1) Substitute back into the equation, and (2) make sure both side lengths are positive. Here: AB = 8 and AC = 5+2 = 7, both positive ✓

Parallelogram Perimeter with Variable Sides

A parallelogram is shown below.

AB = 8

AC = X+2

The perimeter of the parallelogram is 30.

Calculate X.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Opposite sides are equal in a parallelogram

00:10 They are also a pair of opposite sides, therefore equal

00:20 The perimeter of the parallelogram equals the sum of its sides

00:33 Let's substitute appropriate values and solve for X

00:53 Let's group the numbers to one factor, and X to one factor

01:04 Isolate X

01:17 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

A parallelogram is shown below.

AB = 8

AC = X+2

The perimeter of the parallelogram is 30.

Calculate X.

Step-by-step solution

The problem involves finding the value of $X$ in a parallelogram with sides given and a specified perimeter. We will use the formula for the perimeter of a parallelogram.

The formula for the perimeter $P$ of a parallelogram is:

P = 2(a + b)

Given that:

One side $AB = a = 8$ .
The other side $AC = b = X + 2$ .
The perimeter $P = 30$ .

Substitute the given values into the perimeter formula:

2(8 + (X + 2)) = 30

Simplify the expression inside the parentheses:

8 + X + 2 = X + 10

Now the equation becomes:

2(X + 10) = 30

Divide both sides by 2:

X + 10 = 15

Subtract 10 from both sides to solve for $X$ :

X = 15 - 10

Thus:

X = 5

The value of $X$ is therefore $\textbf{5}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Property: Opposite sides of a parallelogram are equal in length
Formula: Perimeter = 2(side₁ + side₂) = 2(8 + (X+2))
Check: Substitute X=5: 2(8 + 7) = 2(15) = 30 ✓

Common Mistakes

Avoid these frequent errors

Adding all four sides separately instead of using the parallelogram property
Don't write 8 + (X+2) + 8 + (X+2) = 30 then solve manually! This makes the algebra messier and increases error chances. Always use the parallelogram property: Perimeter = 2(side₁ + side₂) to simplify first.

Practice Quiz

Test your knowledge with interactive questions

Given the parallelogram:

Calculate the perimeter of the parallelogram.

FAQ

Everything you need to know about this question

Why do we only need two side lengths for a parallelogram's perimeter?

In a parallelogram, opposite sides are always equal. So if AB = 8, then the opposite side CD also equals 8. If AC = X+2, then the opposite side BD also equals X+2.

How do I know which sides are opposite to each other?

Look at the vertices in order: A-B-D-C. Side AB is opposite to side CD, and side AC is opposite to side BD. Opposite sides never touch each other - they're parallel!

What if I get a negative value for X?

Check your algebra! In geometry problems, side lengths must be positive numbers. If you get X = -3, then AC = X+2 = -1, which doesn't make sense for a length.

Can I solve this without the perimeter formula?

Yes, but it's harder! You could write out all four sides: $8 + (X+2) + 8 + (X+2) = 30$ . The formula $P = 2(a + b)$ just makes it cleaner.

How do I verify my answer makes sense?

Two checks: (1) Substitute back into the equation, and (2) make sure both side lengths are positive. Here: AB = 8 and AC = 5+2 = 7, both positive ✓

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