Solve for X: Parallelogram with Perimeter 20 and Side Ratio 1:3

Q: Why do we multiply by 2 in the perimeter formula?

A parallelogram has four sides, but only two different lengths. Since opposite sides are equal, we have two sides of length x and two sides of length 3x, so P = 2x + 2(3x) = 2(x + 3x).

Q: What if I set up the equation wrong?

Always check your setup first! Make sure you're using P = 2(sum of different side lengths), not just adding the given expressions. Then solve step by step.

Perimeter Equations with Variable Expressions

Calculate X in the parallelogram shown below.

P = Perimeter

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 Opposite sides are equal in a parallelogram

00:12 The perimeter of the parallelogram equals the sum of its sides

00:30 Substitute appropriate values and solve for X

00:37 Isolate X

00:46 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

Calculate X in the parallelogram shown below.

P = Perimeter

Step-by-step solution

To calculate $x$ in the given parallelogram, we begin by noting the given perimeter formula for the parallelogram, $P = 2(a + b)$ .

The two sides, as given, are $a = x$ and $b = 3x$ . Substituting these into the formula gives:

$P = 2(x + 3x)$

We know the perimeter $P = 20$ , therefore:

$2(x + 3x) = 20$

Simplifying inside the parentheses, $x + 3x = 4x$ , we rewrite the equation as:

$2 \times 4x = 20$

Which gives:

$8x = 20$

To solve for $x$ , divide both sides by 8:

$x = \frac{20}{8}$

Simplifying gives:

$x = \frac{5}{2}$

Or, $x = 2.5$ .

Thus, the value of $x$ is $\boxed{2\frac{1}{2}}$ .

Final Answer

$2\frac{1}{2}$

Key Points to Remember

Essential concepts to master this topic

Perimeter Formula: For parallelograms, P = 2(length + width)
Technique: Substitute x and 3x into P = 2(x + 3x) = 20
Check: Verify 2.5 works: 2(2.5 + 7.5) = 2(10) = 20 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply by 2 in perimeter formula
Don't use P = x + 3x = 20, getting x = 5! This ignores that a parallelogram has two pairs of opposite sides. Always use P = 2(x + 3x) to account for all four sides.

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 multiply by 2 in the perimeter formula?

A parallelogram has four sides, but only two different lengths. Since opposite sides are equal, we have two sides of length x and two sides of length 3x, so P = 2x + 2(3x) = 2(x + 3x).

How do I know which sides are x and which are 3x?

Look at the diagram! The labels show one pair of opposite sides is x and the other pair is 3x. In parallelograms, opposite sides are always equal.

Can x be a fraction or decimal?

Absolutely! Many geometry problems have fractional answers. $x = 2\frac{1}{2}$ or 2.5 is perfectly valid for a side length.

What if I set up the equation wrong?

Always check your setup first! Make sure you're using P = 2(sum of different side lengths), not just adding the given expressions. Then solve step by step.

How can I verify my answer is correct?

Substitute back: if x = 2.5, then the sides are 2.5 and 7.5. Check: P = 2(2.5 + 7.5) = 2(10) = 20 ✓. Always verify with the original perimeter!

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